Unified Bayesian conditional autoregressive risk measures using the skew exponential power distribution
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ORIGINAL PAPER
Unified Bayesian conditional autoregressive risk measures using the skew exponential power distribution Marco Bottone1
•
Lea Petrella2 • Mauro Bernardi3
Accepted: 3 November 2020 Ó Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Conditional Autoregressive Value-at-Risk and Conditional Autoregressive Expectile have become two popular approaches for direct measurement of market risk. Since their introduction several improvements both in the Bayesian and in the classical framework have been proposed to better account for asymmetry and local non-linearity. Here we propose a unified Bayesian Conditional Autoregressive Risk Measures approach by using the Skew Exponential Power distribution. Further, we extend the proposed models using a semiparametric P-Spline approximation answering for a flexible way to consider the presence of non-linearity. To make the statistical inference we adapt the MCMC algorithm proposed in Bernardi et al. (2018) to our case. The effectiveness of the whole approach is demonstrated using real data on daily return of five stock market indices. Keywords Bayesian quantile regression Skew exponential power Risk measure Adaptive-MCMC CAViaR model CARE model
1 Introduction After the recent financial crisis an accurate risk measurement is a primary need for financial institutions and investors. Within the instruments for market risk measurement, Value-at-Risk (VaR) (Jorion 2007) and Expected Shortfall (ES) (Artzner et al. 1999) are certainly the most popular and used approaches. VaR answers the question on what is the maximum potential loss that will be exceeded & Marco Bottone [email protected] 1
DG for Economics, Statistics and Research, Banca d’Italia, Via Nazionale 91, 00184 Rome, Italy
2
Department of Methods and Models for Economics, Territory and Finance, Sapienza University of Rome, Rome, Italy
3
Department of Statistical Sciences, University of Padua, Padua, Italy
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with a certain probability in the next days. It can be simply understood as a specific (say s) conditional quantile of the portfolio returns given the current information, i.e., PðYt \ VaRt j F t Þ ¼ s, where Yt and F t denote the return of a portfolio and the information set available at time t, respectively, while s 2 ð0; 1Þ denotes the quantile confidence level associated with the VaR. Even though it is widely used among financial institutions VaR has been criticized because of the absence of the sub-additivity property, namely, it does not guarantee that a diversified portfolio is less risky than a concentrated one. In addition VaR gives no information regarding possible exceedances beyond the quantile which may be quite important in evaluating risks. Artzner et al. (1999) first recognized the lack of coherency of the VaR and proposed the ES as an alternative coherent risk measure which gives more information about the returns’ distribution in the tails. In particular the ES is defined as the
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