Abstract
Value at Risk (VaR) has become a crucial measure for decision making in risk management over the last thirty years and many estimation methodologies address the finding of the best performing measure at taking into account unremovable uncertainty of real financial markets. One possible and promising way to include uncertainty is to refer to the mathematics of fuzzy numbers and to its rigorous methodologies which offer flexible ways to read and to interpret properties of real data which may arise in many areas. The paper aims to show the effectiveness of two distinguished models to account for uncertainty in VaR computation; initially, following a non parametric approach, we apply the Fuzzy-transform approximation function to smooth data by capturing fundamental patterns before computing VaR. As a second model, we apply the Average Cumulative Function (ACF) to deduce the quantile function at point p as the potential loss VaRp for a fixed time horizon for the 100p% of the values. In both cases a comparison is conducted with respect to the identification of VaR through historical simulation: twelve years of daily S&P500 index returns are considered and a back testing procedure is applied to verify the number of bad VaR forecasting in each methodology. Despite the preliminary nature of the research, we point out that VaR estimation, when modelling uncertainty through fuzzy numbers, outperforms the traditional VaR in the sense that it is the closest to the right amount of capital to allocate in order to cover future losses in normal market conditions.
Highlights
In 1996 the Basel Committee approved the use of proprietary Value at Risk (VaR) measures for calculating the market risk component of bank capital requirements; from that year, the scientific literature grew dramatically in order to identify the best performing way to measure VaR.Jorion in [1] defines VaR as the measure that is the worst expected loss over a given horizon under normal market conditions at a given level of confidence
The second approach addresses non parametric methods and, as in [19], we propose a method to estimate quantiles through a nonparametric estimates of the cumulative distribution function deduced by using the Average Cumulative Function (ACF) which plays the role of the double kernel smoothing in the mentioned paper
The non parametric smoothing methods based on F-transform can weaken these distortions as we show in some simple experiments
Summary
In 1996 the Basel Committee approved the use of proprietary Value at Risk (VaR) measures for calculating the market risk component of bank capital requirements; from that year, the scientific literature grew dramatically in order to identify the best performing way to measure VaR. We believe that the modeling of uncertainty through fuzzy logic in decision making and risk management deserves in-depth analysis; just to mention some of our contributions, we developed the rigorous use of the extension principle for fuzzy-valued functions in [15] where we show that fuzzy financial option prices can capture the unavoidable uncertainty of several stylized facts in real markets and the subjective believes of the investor It is a matter of interest the use of VaR as a factor fixing decision-making believes for risk-averse investors, for example in [16], the process of recovering investment opportunities with projects that have been rejected when applying the criterion of the Value-at-Risk method, is studied.
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