Applications of Copulas for the Calculation of Value-at-Risk

Estimation of risk measures in energy portfolios using modern copula techniques

“…Therefore, the present paper simulates the Value-at-Risk and Expected Shortfall for both models and compares the results for different portfolio weights and certain levels of confidence (Rank and Siegl, 2002). As suspected and confirmed by the backtesting the Gumbel model slightly outperforms the t-copula.…”

“…Following Rank and Siegl (2002) we first generate n pairs (u, v) of observations of U (0, 1) distributed random variables U and V whose joint distribution function is either given by the t-copula (θ T = 0.895, ν T = 5.313) from Section 5.1 or the survival copula of the Gumbel copula (θ T = 2.961) from Section 5.2. The simulation is carried out using the functionality of and Yan (2007).…”

“…Figure 7 shows the out of sample one day-ahead simulated portfolio (ω 1 = ω 2 = 0.5) Value-at-Risk for a given level of confidence α = 95% for the estimated Gumbel model in combination with the realised (observed) returns (quod vide Palaro and Hotta, 2006). To be able to judge the effectiveness of the applied Value-at-Risk model, we compare the simulated (predicted) and empirical number of outliers, where the actual loss exceeds the Value-at-Risk (Rank and Siegl, 2002). Consequently, from a risk management point of view, this backtesting approach is easily comprehensible and allows for the required transparency.…”

“…Table 4 summarises the results of the backtesting for the set of scenarios w 1,i (ω 1,1 = 0.50, ω 1,2 = 0.25, ω 1,3 = 0.75) with a given level of confidence α. Following Rank and Siegl (2002) we define the prediction error as the absolute difference between the relative number of outliers 1 −α and the predicted relative number 1 − α. While the average over the portfolios uses equal weights, the average over the level of confidence α emphasises the tails by a weighting scheme β i (β 1 = 1, β 2 = 2, β 3 = 5).…”

Portfolio value‐at‐risk estimation for spot chartering decisions under changing trade patterns: A copula approach

“…In order to evaluate the effectiveness of the applied VaR model, we compare the difference between the estimated number of violations, using different VaR estimation models, and the expected number of violations given a predefined confidence interval where the actual loss exceeds the VaR (Rank & Siegl, 2002). These backtesting results are summarized in Table 5.…”