Analysis and Backtesting

The backtesting of simulated return and price time-series shows that our approach is able to reproduce some stylized facts (Andersen and Davis 2009). There is an auto­correlation in the absolute values of simulated returns (Malmsten and Terasvirta 2004), but it decays very fast (Fig. 7).


Table 3 Estimated risk-metrics on scenarios

Basic scenario

Stress scenario “GMKN-40 %”

Maximum loss

39.90 %

42.30 %

Maximum profit

59.40 %

57 %

90 % VaR


-16.30 %

95 % VaR


-20.20 %

99 % VaR

-26.80 %

-29.20 %

90 % ES


-23.50 %

95 % ES

-26.90 %

-29.30 %

99 % ES

-39.90 %

-42.30 %

QQ-charts show that the distribution of simulated returns differs from normal and (Fig. 8) and demonstrates heavy-tails fairly close to the distribution of historical returns (Fig. 9).


Fig. 8 QQ-charts for the distribution of simulated returns in comparison with normal distribution



Fig. 9 QQ-charts for the distribution of simulated returns in comparison with the distribution of historical returns



We propose a new approach for stress-testing of a given investment portfolio based on the application of the GARCH model with a particular specification for the model’s error together with the copula’s description of risk factors dependency structure. This method can be backtested by the reproduction of stylized facts known for returns and price time-series:





Подпись: • There is autocorrelation in simulated return time-series, but it decays fairly fast; • The distribution of simulated returns differs from normal and has heavy tails pretty close to the distribution of historical returns; • There is a volatility clustering effect in the simulated returns; • By using the copula for a dependency structure description it is possible to catch various and complicated changes in dependencies between the risk factors. The model allows us to simulate the returns of the portfolio according to the variations in risk factors for use for profit-loss distribution estimation, as well as market risk measurement under stress conditions.


Andersen, T. G., & Davis, R. A. (2009). Handbook of financial time series. Berlin: Springer.

Basel Committee on Banking Supervision. (2009). Principles for sound stress testing practices and supervision. Basel consultative documents. www. bis. org/publ/bcbs155.pdf.

Charpentier, A. (2006). The estimation of copulas: theory and practice. http://perso. univ-rennes1. fr/arthur. charpentier/chapter-book-copula-density-estimation. pdf.

Genest, C., Gendron, M., & Bourdeau-Brien, M. (2009). The advent of copulas in finance. The European Journal of Finance, 15, 609-618.

Malmsten, H., & Terasvirta, T. (2004). Stylized facts of financial time series and three pop­ular models of volatility. http://ljsavage. wharton. upenn. edu/~steele/Resources/FTSResources/ StylizedFacts/MalmstenTerasvirta04.pdf.

Posedel, P. (2005). Properties and estimation of GARCH(1,1) model. http://www. ressources- actuarielles. net/EXT/ISFA/1226.nsf/0/73d982a644ea2f6dc1257609006edb99/$FILE/posedel. pdf.

Sorge, M. (2004). Stress-testing financial systems: an overview of current methodologies. http:// www. bis. org/publ/work165.pdf.

Schimdt, T. (2006). Coping with copulas. http://www. math. uni-leipzig. de/~tschmidt/TSchmidt_ Copulas. pdf.

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