Modelos de risco de mercado com fat tail : análise empírica de value at risk and expected shortfall para ativos financeiros brasileiros

O objetivo deste trabalho foi mostrar modelagens alternativas à tradicional maneira de se apurar o risco de mercado para ativos financeiros brasileiros. Procurou-se cobrir o máximo possível de fatores de risco existentes no Brasil; para tanto utilizamos as principais proxies para instrumentos de Renda Fixa. Em momentos de volatilidade, o gerenciamento de risco de mercado é bastante criticado por trabalhar dentro de modelagens fundamentadas na distribuição normal. Aqui reside a maior contribuição do VaR e também a maior crítica a ele. Adicionado a isso, temos um mercado caracterizado pela extrema iliquidez no mercado secundário até mesmo em certos tipos de títulos públicos federais. O primeiro passo foi fazer um levantamento da produção acadêmica sobre o tema, seja no Brasil ou no mundo. Para a nossa surpresa, pouco, no nosso país, tem se falado em distribuições estáveis aplicadas ao mercado financeiro, seja em gerenciamento de risco, precificação de opções ou administração de carteiras. Após essa etapa, passamos a seleção das variáveis a serem utilizadas buscando cobrir uma grande parte dos ativos financeiros brasileiros. Assim, deveríamos identificar a presença ou não da condição de normalidade para, aí sim, realizarmos as modelagens das medidas de risco, VaR e ES, para os ativos escolhidos, As condições teóricas e práticas estavam criadas: demanda de mercado (crítica ao método gausiano bastante difundido), ampla cobertura de ativos (apesar do eventual questionamento da liquidez), experiência acadêmica e conhecimento internacional (por meio de detalhado e criterioso estudo da produção sobre o tema nos principais meios). Analisou-se, desta forma, quatro principais abordagens para o cálculo de medidas de risco sendo elas coerentes (ES) ou não (VaR). É importante mencionar que se trata de um trabalho que poderá servir de insumo inicial para trabalhos mais grandiosos, por exemplo, aqueles que incorporarem vários ativos dentro de uma carteira de riscos lineares ou, até mesmo, para ativos que apresentem risco não-direcionais.

Value at risk e expectes shortfall : medidas de risco e suas propriedades : um estudo empírico para o mercado brasileiro

Value at Risk (VaR) e Expected Shortfall (ES) são modelos quantitativos para mensuração do risco de mercado em carteiras de ativos financeiros. O propósito deste trabalho é avaliar os resultados de tais modelos para ativos negociados no mercado brasileiro através de quatro metodologias de backtesting - Basel Traffic Light Test, Teste de Kupiec, Teste de Christoffersen e Teste de McNeil e Frey – abrangendo períodos de crise financeira doméstica (2002) e internacional (2008). O modelo de VaR aqui apresentado utilizou duas abordagens – Paramétrica Normal, onde se assume que a distribuição dos retornos dos ativos segue uma Normal, e Simulação Histórica, onde não há hipótese a respeito da distribuição dos retornos dos ativos, porém assume-se que os mesmos são independentes e identicamente distribuídos. Também foram avaliados os resultados do VaR com a expansão de Cornish-Fisher, a qual visa aproximar a distribuição empírica a uma distribuição Normal utilizando os valores de curtose e assimetria para tal. Outra característica observada foi a propriedade de coerência, a qual avalia se a medida de risco obedece a quatro axiomas básicos – monotonicidade, invariância sob translações, homogeneidade e subaditividade. O VaR não é considerado uma medida de risco coerente, pois não apresenta a característica de subaditividade em todos os casos. Por outro lado o ES obedece aos quatro axiomas, considerado assim uma medida coerente. O modelo de ES foi avaliado segundo a abordagem Paramétrica Normal. Neste trabalho também se verificou através dos backtests, o quanto a propriedade de coerência de uma medida de risco melhora sua precisão.

Higher order elicitability and Osband’s principle

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance, this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the “correct” quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.

Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting

In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk.

An asymptotic test for the Conditional Value-at-Risk

Conditional Value-at-Risk (equivalent to the Expected Shortfall, Tail Value-at-Risk and Tail Conditional Expectation in the case of continuous probability distributions) is an increasingly popular risk measure in the fields of actuarial science, banking and finance, and arguably a more suitable alternative to the currently widespread Value-at-Risk. In my paper, I present a brief literature survey, and propose a statistical test of the location of the CVaR, which may be applied by practising actuaries to test whether CVaR-based capital levels are in line with observed data. Finally, I conclude with numerical experiments and some questions for future research.

Forecasting time-varying value-at-risk

In this thesis we are interested in financial risk and the instrument we want to use is Value-at-Risk (VaR). VaR is the maximum loss over a given period of time at a given confidence level. Many definitions of VaR exist and some will be introduced throughout this thesis. There two main ways to measure risk and VaR: through volatility and through percentiles. Large volatility in financial returns implies greater probability of large losses, but also larger probability of large profits. Percentiles describe tail behaviour. The estimation of VaR is a complex task. It is important to know the main characteristics of financial data to choose the best model. The existing literature is very wide, maybe controversial, but helpful in drawing a picture of the problem. It is commonly recognised that financial data are characterised by heavy tails, time-varying volatility, asymmetric response to bad and good news, and skewness. Ignoring any of these features can lead to underestimating VaR with a possible ultimate consequence being the default of the protagonist (firm, bank or investor). In recent years, skewness has attracted special attention. An open problem is the detection and modelling of time-varying skewness. Is skewness constant or there is some significant variability which in turn can affect the estimation of VaR? This thesis aims to answer this question and to open the way to a new approach to model simultaneously time-varying volatility (conditional variance) and skewness. The new tools are modifications of the Generalised Lambda Distributions (GLDs). They are four-parameter distributions, which allow the first four moments to be modelled nearly independently: in particular we are interested in what we will call para-moments, i.e., mean, variance, skewness and kurtosis. The GLDs will be used in two different ways. Firstly, semi-parametrically, we consider a moving window to estimate the parameters and calculate the percentiles of the GLDs. Secondly, parametrically, we attempt to extend the GLDs to include time-varying dependence in the parameters. We used the local linear regression to estimate semi-parametrically conditional mean and conditional variance. The method is not efficient enough to capture all the dependence structure in the three indices —ASX 200, S&P 500 and FT 30—, however it provides an idea of the DGP underlying the process and helps choosing a good technique to model the data. We find that GLDs suggest that moments up to the fourth order do not always exist, there existence appears to vary over time. This is a very important finding, considering that past papers (see for example Bali et al., 2008; Hashmi and Tay, 2007; Lanne and Pentti, 2007) modelled time-varying skewness, implicitly assuming the existence of the third moment. However, the GLDs suggest that mean, variance, skewness and in general the conditional distribution vary over time, as already suggested by the existing literature. The GLDs give good results in estimating VaR on three real indices, ASX 200, S&P 500 and FT 30, with results very similar to the results provided by historical simulation.

Diversification and value-at-risk

A pervasive and puzzling feature of banks’ Value-at-Risk (VaR) is its abnormally high level, which leads to excessive regulatory capital. A possible explanation for the tendency of commercial banks to overstate their VaR is that they incompletely account for the diversification effect among broad risk categories (e.g., equity, interest rate, commodity, credit spread, and foreign exchange). By underestimating the diversification effect, bank’s proprietary VaR models produce overly prudent market risk assessments. In this paper, we examine empirically the validity of this hypothesis using actual VaR data from major US commercial banks. In contrast to the VaR diversification hypothesis, we find that US banks show no sign of systematic underestimation of the diversification effect. In particular, diversification effects used by banks is very close to (and quite often larger than) our empirical diversification estimates. A direct implication of this finding is that individual VaRs for each broad risk category, just like aggregate VaRs, are biased risk assessments.

The level and quality of Value-at-Risk disclosure by commercial banks

In this paper we study both the level of Value-at-Risk (VaR) disclosure and the accuracy of the disclosed VaR figures for a sample of US and international commercial banks. To measure the level of VaR disclosures, we develop a VaR Disclosure Index that captures many different facets of market risk disclosure. Using panel data over the period 1996–2005, we find an overall upward trend in the quantity of information released to the public. We also find that Historical Simulation is by far the most popular VaR method. We assess the accuracy of VaR figures by studying the number of VaR exceedances and whether actual daily VaRs contain information about the volatility of subsequent trading revenues. Unlike the level of VaR disclosure, the quality of VaR disclosure shows no sign of improvement over time. We find that VaR computed using Historical Simulation contains very little information about future volatility.

An evaluation of the effectiveness of Value-at-Risk (VaR) models for Australian banks under Basel III

This study compares Value-at-Risk (VaR) measures for Australian banks over a period that includes the Global Financial Crisis (GFC) to determine whether the methodology and parameter selection are important for capital adequacy holdings that will ultimately support a bank in a crisis period. VaR methodology promoted under Basel II was largely criticised during the GFC for its failure to capture downside risk. However, results from this study indicate that 1-year parametric and historical models produce better measures of VaR than models with longer time frames. VaR estimates produced using Monte Carlo simulations show a high percentage of violations but with lower average magnitude of a violation when they occur. VaR estimates produced by the ARMA GARCH model also show a relatively high percentage of violations, however, the average magnitude of a violation is quite low. Our findings support the design of the revised Basel II VaR methodology which has also been adopted under Basel III.

An Empirical Investigation of Value-at-Risk in Long and Short Trading Positions

This paper uses the Value-at-Risk approach to define the risk in both long and short trading positions. The investigation is done on some major market indices(Japanese, UK, German and US). The performance of models that takes into account skewness and fat-tails are compared to symmetric models in relation to both the specific model for estimating the variance, and the distribution of the variance estimate used as input in the VaR estimation. The results indicate that more flexible models not necessarily perform better in predicting the VaR forecast; the reason for this is most probably the complexity of these models. A general result is that different methods for estimating the variance are needed for different confidence levels of the VaR, and for the different indices. Also, different models are to be used for the left respectively the right tail of the distribution.