International Journal of Mathematical, Engineering and Management Sciences Vol. 4, No. 3, 542–566, 2019 https://dx.doi.org/10.33889/IJMEMS.2019.4.3-044 542 Post Model Correction in Risk Analysis and Management G.-J. Siouris, D. Skilogianni, A. Karagrigoriou * Lab of Statistics and Data Analysis Department of Statistics and Actuarial-Financial Mathematics University of the Aegean, Greece * Corresponding author: alex.karagrigoriou@aegean.gr (Received January 19, 2019; Accepted February 21, 2019) Abstract This work focuses on Value at Risk (VaR) and Expected Shortfall (ES) in conjunction with the so called, low price effect. In order to improve forecasts of risk measures like VaR or ES when low price effect is present, we propose the low price correction which does not involve additional parameters and instead of returns it relies on asset prices. The forecasting ability of the proposed methodology is measured by appropriately adjusted popular evaluation measures, like MSE and MAPE as well as by backtesting methods. For illustrative and comparative purposes a real example from the Athens Stock Exchange as well as a number of penny stocks from Nasdaq, NYSE and NYSE MKT are fully examined. The proposed technique is always applicable, but its superiority and effectiveness is evident in extreme economic scenarios and severe stock collapses. The proposed methodology that pays attention not only to the asset return but also to the asset price, provides sufficient evidence that prices could contain important information which could if taken under consideration, results in improved forecasts of risk estimation. Keywords- EWMA, ARCH, GARCH, APARCH, FIGARCH, Expected shortfall, VaR, PVaR, Violation ratios; Normalised shortfall, EPS, Leverage effect, Low price effect, Low price correction, Backtesting. 1. Introduction The quantification of risk is an important issue in finance that becomes even more important during periods of financial crises. Risk measures have been proposed and used, over the years, to evaluate the overall risk exposure for the purpose of financial supervision including internal control and banking supervision. Value at Risk is the most popular of such measures primarily due to its simplicity. Furthermore, VaR is easily validated and backtested by simply comparing predicted and actual values. On the other hand, VaR is not a formal risk measure since it fails to fulfill the axiom of sub-additivity according to which portfolio diversification leads to risk reduction. An additional drawback of VaR is its inability to capture tail risk. The heavy tail property refers to instances with large positive or negative returns unlikely to be observed in bell-shaped symmetric returns. Simply speaking, measures like VaR do not provide information about the magnitude of the loss beyond a threshold level. Fat tails is one of the financial concepts that appear frequently and is considered as one of the sources of inadequate and often inaccurate estimation of the total risk exposure (Demírgüç-Kunt and Levine, 1996; Cont, 2001). The introduction of the Expected Shortfall as a new risk measure that possesses the sub-additivity property and measures the loss in the tail, came naturally as a response to the criticism of VaR (Artzner et.al., 1999; Acerbi and Tasche, 2002). The estimation of volatility is the main financial characteristic associated with risk analysis and management. In order to forecast simultaneously returns and volatility, different models have been developed over the years which include, among others Moving average models (MA), Exponentially weighted moving average models (EWMA), Autoregressive conditional heteroskedasticity models (ARCH), Generalized ARCH (GARCH) models and their extensions,