Sensitivity Analysis of Two-Step Multinomial Backtests for Evaluating Value-at-Risk

Document Type : Research Paper


1 Assistant Prof., Department of Financial Engineering, Faculty of Industrial Engineering & Systems, Tarbiat Modares University, Tehran, Iran.

2 M.Sc., Department of Finance, Faculty of Management and Finance, Khatam University, Tehran, Iran.


Objective: Nowadays, the measurement of the risk of the marketplace has a significant effect on investments; however, the inadequate evaluation of this risk will cause a financial crisis and possible bankruptcy. One of the typical approaches to measure this risk is the probability-based risk measurement method, known as Value-at-Risk (VaR), for estimating and backtesting of which there are various methods. The purpose of this paper is to put forward a comprehensive test for backtesting and analyzing the sensitivity of VaR based on the number of samples (n) and confidence levels (N).
Methods: First, the VaR of Tehran Stock Exchange data was estimated by applying GARCH-Copula, DCC, and EVT. Next, by using the multinomial backtesting in two steps the accuracy of VaR estimation and ranked the models were tested. Thereafter, considering the number of samples (n) and the confidence levels (N), the sensitive analysis of the backtesting result demonstrated the accuracy of the estimated VaR by selecting the most appropriate parameters.
Results: Sensitive analysis findings indicated that in all three models, increasing the parameter "N" will result in an increase in the error rate. On the other hand, sensitive analysis of parameter "n" proved that its value depends on the technique used to estimate VaR, but generally, any increase in it leads to validation of VaR estimation models. The results also showed that according to the EVT method, at least 29% of the data is required to be used as a test sample in VaR estimation; however, the amount is equal to 22% in the DCC and GARCH-Copula methods.
Conclusion: The result of the sensitivity analysis indicated that the reliability of different estimating VaR techniques relies on "n" and "N" parameters and different amounts of these two parameters can generate inaccurate and uncertain outcomes for each model. In addition, ranking these methods by using the loss function, GARCH-Copula, EVT and DCC methods ranked first to third, respectively.


Angelidis, T., & Degiannakis, S. A. (2018). Backtesting VaR Models: A Τwo-Stage Procedure. Available at SSRN 3259849.
Bellini, F., Negri, I., & Pyatkova, M. (2019). Backtesting VaR and expectiles with realized scores. Statistical Methods & Applications28(1), 119-142.
Braione, M., & Scholtes, N. K. (2016). Forecasting value-at-risk under different distributional assumptions. Econometrics4(1), 3.
Bücher, A., Posch, P. N., & Schmidtke, P. (2020). Using the Extremal Index for Value-at-Risk Backtesting. Journal of Financial Econometrics, 18(3), 556-584.
Christoffersen, P., & Pelletier, D. (1998). Backtesting value-at-risk: A duration-based approach. Journal of Financial Econometrics2(1), 84-108.
De Haan, L., & Ferreira, A. (2006). Extreme value theory: an introduction. Springer Science & Business Media.
Engle, R. (2002). Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business & Economic Statistics20(3), 339-350.
Gkillas, K., & Katsiampa, P. (2018). An application of extreme value theory to cryptocurrencies. Economics Letters164, 109-111.
Gorji, M. & Sajjad, R. (2016). Estimation of multi-period VaR based on the simulation and parametric methods. Financial Research Journal, 18(1), 167-184. (in Persian)
Kashi, M., Hosseyni, H., Niazkhani, A., & Abdollahi, A. (2016). VaR modeling and backtesting of short and long positions according to in Sample and out of Sample: application of family models Fractionally Integrated GARCH. Journal of Financial Engineering and Portfolio Management, 7(29), 1-24. (in Persian)
Keshavarz Hadad. Q., & Heyrani. M. (2015). Estimation of Value at Risk in the Presence of Dependence Structure in Financial Returns: A Copula Based Approach. Journal of Economic Research, 49(4), 869-902. (in Persian)
Kratz, M., Lok, Y. H., & McNeil, A. J. (2018). Multinomial VaR Backtests: A simple implicit approach to backtesting expected shortfall. Journal of Banking & Finance88, 393-407.
Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. The Journal of Derivatives3(2), 73-84.
Patton, A. J. (2004). On the out-of-sample importance of skewness and asymmetric  dependence for asset allocation. Journal of Financial Econometrics, 2(1), 130-168.
Raei, R. Basakha, H., & Mahdikhah, H. (2020). Equity Portfolio Optimization Using Mean-CVaR Method Considering Symmetric and Asymmetric Autoregressive Conditional Heteroscedasticity. Financial Research Journal, 22(2), 149-159. (in Persian)
Rostami Noroozabad, M. Shojaei, A. Khezri, M., & Rostami Noroozabad, S. (2015). Estimation of value at risk of return in Tehran Stock Exchange using wavelet analysis. Financial Research Journal, 17(1), 59-82. (in Persian)
Saranj, A., & Nourahmadi, M. (2016). Estimating of value at risk and expected shortfall by using conditional extreme value approach in Tehran Securities Exchange. Financial Research Journal, 18(3), 437-460. (in Persian)
Shekari, S. (2018). Predicting VaR based on time-varying Copula. Master Thesis, Kharazmi University. (in Persian)
Taleblou, R., & Davoudi, M. (2018). Estimating the optimal investment portfolio by using Value at Risk (VaR) and Expected Shortfall (ES): GARCH-EVT-Copula approach. Journal of Applied Economics Studies, 18(71), 91-125. (in Persian)
Taleblou, R., & Davoudi, M. (2020). Estimating of value at risk: DCC-GARCH-Copula Method. Iranian Journal of Economic Research, 25(82), 43-82. (in Persian)
Wied, D., Weiß, G. N., & Ziggel, D. (2016). Evaluating Value-at-Risk forecasts: A new set of multivariate backtests. Journal of Banking & Finance72, 121-132.