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Numerical comparison of multivariate models to forecasting risk measures

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Abstract

We evaluated the performance of multivariate models for forecasting Value at Risk (VaR), Expected Shortfall (ES), and Expectile Value at Risk (EvaR). We used Historical Simulation (HS), Dynamic Conditional Correlation-Generalized Autoregressive Conditional Heteroskedastic (DCC-GARCH) and copula methods: Regular copulas, Vine copulas, and Nested Archimedean copulas (NAC). We assessed the performance of the models using Monte Carlo simulations, considering different scenarios, regarding the marginal distributions, correlation, and number of portfolio assets. Numerical results evidenced the accuracy forecasting risk measures are associated with marginal distributions. For a data-generating process where the marginal distribution is Gaussian, Regular and Vine copulas demonstrated better performance. For data generated with Student’s t distribution, we verified better performance by NAC. In addition, we identified the superiority of copula methods over HS and DCC-GARCH, which reduces the model risk.

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Notes

  1. A coherent risk measure is a risk measure that satisfies four axioms: translation invariance, subadditivity, positive homogeneity, and monotonicity (Artzner et al. 1999).

  2. Risk measures are called elicitable when the verification and comparison of competing estimation procedures is possible (Ziegel 2014). See also Ziegel (2014) and Bellini and Bernardino (2017) for a detailed discussion referent to elicitability for risk measures.

  3. A bivariate distribution function C(u, v) with marginals \(F_1\) and \(F_2\) is generated by an Archimedean copula if it can be given by \(C(u,v) = \varphi ^{-1}[\varphi (F_1(u)) +\varphi (F_2(v))]\), where \(\varphi\) is a function \(\varphi : I \rightarrow \mathbb {R}^{*+}\), continuous, decreasing, convex and such that \(\phi (1) = 0\) (Cherubini et al. 2004).

  4. We refer to the Law of Large Numbers and Central Limit Theorem (CLT) in which Monte Carlo simulations are based for a better understanding.

  5. This research focused on comparing the performance of traditional multivariate models, such as HS and DCC-GARCH, with copulas. We did not use models such as multivariate quantile regression and multivariate Extreme Value Theory.

  6. Model risk refers to errors in modelling assumptions that introduce errors in risk measurement (Glasserman and Xu 2014).

  7. Revisions to the Basel II Market Risk Framework require financial institutions quantify model risk (Basel Committee on Banking Supervision 2009).

  8. To validate the choice of a D-vine, we compared the estimated model with its counterpart through the test presented by Clarke (2007). We have not included the results here, but they are available upon request.

  9. The elements of univariate GARCH were obtained through the structure given in (12). The model GARCH(P,Q) can be described in the following manner: \(X_t = \mu _t + \sigma _tz_t, \quad \sigma _t^2 = \omega + \sum _{p = 1}^{P}a_{p}\epsilon ^2_{t-p} + \sum _{q = 1}^{Q}b_{q}\sigma ^2_{t-q}\), where \(\mu _t\) represents the expectancy, \(z_t\) is independent and identically distributed (i.i.d.) with zero mean and unit variance, \(\omega\) is the constant, a the component ARCH, and b represents the component GARCH. One can use other generalizations of the univariate GARCH model.

  10. Studies show 1000 observations (four years of daily data) is a good sample size for daily data.

  11. We focused on the 1-day-ahead forecast, because according to the literature, this is the horizon usually used in empirical studies and simulations analysis.

  12. Many works use similar values for the parameter, such as Christoffersen and Gonçalves (2005) and Righi and Ceretta (2015a), because this data-generating process matches the daily returns obtained on the S & P 500 Index.

  13. We generated portfolios with \(2^n\) assets, where \(n = 2, 3, 4\). We have not presented the results of the portfolio with 8 (\(2^3\)) assets, because the results are similar to those of the portfolios with 4 (\(2^2\)) and 16 (\(2^4)\) assets. Portfolios with \(2^5\) or more assets are computationally complicated, due to the copulas approach.

  14. Regarding the parameters estimation, we used quasi-maximum likelihood (Q-ML) estimate.

  15. The distribution used to fit the returns was the same as the scenario considered, to avoid marginal interference (Fantazzini 2009).

  16. In each replicate of Monte Carlo, we generated 10,000 samples \(u_{i,N}\) with the size of 1000 for each asset i. This procedure was repeated 10,000 times.

  17. The method with the better performance would present the smallest value in at least two of these metrics.

  18. In this paper, we considered a model presents a greater model risk if the estimate measure with this model presents greater relative bias.

  19. Institutions that present the same portfolio and use different internal models, approved by the regulator, must hold the same or at least almost the same amount of regulatory capital, which rarely happens, even if institutions use models that pass in backtesting. This problem is referred to as regulatory arbitrage.

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Acknowledgements

We gratefully acknowledge the partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), Brazil.

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  1. School of Business, Federal University of Rio Grande do Sul, Washington Luiz, 855, Porto Alegre, 90010-460, Brazil

    Fernanda Maria Müller & Marcelo Brutti Righi

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  1. Fernanda Maria Müller
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Correspondence to Fernanda Maria Müller.

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Müller, F.M., Righi, M.B. Numerical comparison of multivariate models to forecasting risk measures. Risk Manag 20, 29–50 (2018). https://doi.org/10.1057/s41283-017-0026-8

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