Unconditional Coverage Research Articles

A study on the backtesting and correlation of systemic risk in Chinese financial institutions: an empirical analysis based on MES and Delta CoVaR

Effectively measuring the risk level of financial institutions is the key to preventing and defusing systemic risks. This paper uses the unconditional coverage test method to backtest the systemic risk of 35 listed financial institutions in China, and uses the quantile coherence method to construct a quantile frequency domain correlation network to explore the frequency domain mechanism of systemic risk correlation of financial institutions. The study found that: (1) During the financial market crisis, the Delta CoVaR commonly used systemic risk measurement indicators such as MES, CoVaR, and failed to effectively measure the systemic risk of China's financial institutions. (2) Compared with the short-term correlation layer and the medium-term correlation layer, China's financial institutions have obvious extreme risk linkage effects in the long-term correlation layer.(3) In the short-term, medium-term and long-term correlation layers, small and medium-sized financial institutions such as Ningbo Bank, Industrial Securities and Dongfang Energy all have high correlation importance. Relevant departments should pay attention to preventing the risk of "too correlated to fail" of small and medium-sized financial institutions in my country.

An Empirical Investigation of Value at Risk (VaR) Forecasting Based on Range-Based Conditional Volatility Models

Value at Risk (VaR) is a widely used measure of market risk. Precision in the estimation of volatility leads to accurate VaR forecasts. As volatility is time-varying and has a clustering effect, GARCH class of volatility models is helpful in modeling volatility more precisely. Studies have also shown that range-based volatility estimates are more efficient than traditional models that use only closing prices. Therefore, this study uses the GARCH family of volatility models to model and forecast VaR. The study compares conventional models that use closing prices alone, like GARCH and TARCH, with range-based models like RGARCH and RTARCH models, where the range defined as daily high price minus low price is introduced as an exogenous variable, to explore if the latter provides better predictive accuracy. All the models are back-tested using the Kupiec (1995) unconditional coverage and Christoffersen (1998) conditional coverage tests. The data period in the study ranges from 2003-2021, and we consider five BRICS indices and three major developed economies, namely, the USA, the UK, and Germany. An empirical investigation shows that range-based models do a better job in VaR forecasting as it has more information content than daily closing prices, thereby giving more accurate VaR estimates. The study hopes this finding will greatly help stakeholders like financial institutions, regulators, and practitioners in more effective market risk management.

Forecasting and backtesting systemic risk in the cryptocurrency market

Cryptocurrency has become an increasingly important tool in both portfolio investment and government regulation. As a relatively new asset class, cryptocurrencies are prone to extreme volatility, with the potential for significant downward movements over the short term. This paper uses MES and △CoVaR to forecast the systemic risk in the cryptocurrency market and subsequently tests the validity based on unconditional coverage and independence. The results of this paper show that a DCC-GARCH model performs well in forecasting systemic risk. The paper also shows that Aoen, EOS and Sinacoin are the best forecasters of systemic risk across the 191 cryptocurrencies analysed over the full estimation period. Our findings have important implications for investors and policy-makers with a vested interest in the cryptocurrency market.

Forecasting Value-at-Risk in turbulent stock markets via the local regularity of the price process

A new computational approach based on the pointwise regularity exponent of the price time series is proposed to estimate Value at Risk. The forecasts obtained are compared with those of two largely used methodologies: the variance-covariance method and the exponentially weighted moving average method. Our findings show that in two very turbulent periods of financial markets the forecasts obtained using our algorithm decidedly outperform the two benchmarks, providing more accurate estimates in terms of both unconditional coverage and independence and magnitude of losses.

Modelling the Volatility of Central Bank of Kenya Currency Exchange Rates

In emerging countries, such as Kenya, the foreign exchange market is an important aspect in the economic development of a country. The currency exchange rate market, like the rest of the world's financial markets, has been marked by considerable instabilities over the last decade. The objective of this paper is to model the volatility of the KSH/USD exchange rate prices using and calculate the VaR using the GARCH-EVT model. In particular, this article uses the two-stage GARCH-EVT approach to estimate the value at risk of the Kenyan Shilling against the US dollar., particularly the one-day ahead Value-at-Risk forecast in risk control. The conditional and unconditional coverage test are used to back test the model. We compare the performance of the GARCH-EVT with the daily log returns of key currency in addition to modelling the value at risk in the Kenyan Foreign Exchange market (US dollar) foreign currencies from the period November 2004 – June 2021 for trading days with the exception of holidays and weekends. The mean equation that was best fitting for the data was ARMA (4,2). The optimal GARCH model for the returns of the KSH/USD exchange rate is the GARCH (1,3) with student-t innovations. The results of the backtesting show that GARCH-EVT can be utilized to estimate and forecast VaR at both 5% and 1% level of significance.

Backtesting Marginal Expected Shortfall and Related Systemic Risk Measures

This paper proposes an original approach for backtesting systemic risk measures. This backtesting approach makes it possible to assess the systemic risk measure forecasts used to identify the financial institutions that contribute the most to the overall risk in the financial system. Our procedure is based on simple tests similar to those generally used to backtest the standard market risk measures such as value-at-risk or expected shortfall. We introduce a concept of violation associated with the marginal expected shortfall (MES), and we define unconditional coverage and independence tests for these violations. We can generalize these tests to any MES-based systemic risk measures such as the systemic expected shortfall (SES), the systemic risk measure (SRISK), or the delta conditional value-at-risk ([Formula: see text]CoVaR). We study their asymptotic properties in the presence of estimation risk and investigate their finite sample performance via Monte Carlo simulations. An empirical application to a panel of U.S. financial institutions is conducted to assess the validity of MES, SRISK, and [Formula: see text]CoVaR forecasts issued from a bivariate GARCH model with a dynamic conditional correlation structure. Our results show that this model provides valid forecasts for MES and SRISK when considering a medium-term horizon. Finally, we propose an early warning system indicator for future systemic crises deduced from these backtests. Our indicator quantifies how much is the measurement error issued by a systemic risk forecast at a given point in time which can serve for the early detection of global market reversals.This paper was accepted by Kay Giesecke, finance.

Spectral backtests of forecast distributions with application to risk management

We study a class of backtests for forecast distributions in which the test statistic depends on a spectral transformation that weights exceedance events by a function of the modeled probability level. The weighting scheme is specified by a kernel measure which makes explicit the user’s priorities for model performance. The class of spectral backtests includes tests of unconditional coverage and tests of conditional coverage. We show how the class embeds a wide variety of backtests in the existing literature, and further propose novel variants which are easily implemented, well-sized and have good power. In an empirical application, we backtest forecast distributions for the overnight P&L of ten bank trading portfolios. For some portfolios, test results depend materially on the choice of kernel.

The Improved Value-at-Risk for Heteroscedastic Processes and Their Coverage Probability

A risk measure commonly used in financial risk management, namely, Value-at-Risk (VaR), is studied. In particular, we find a VaR forecast for heteroscedastic processes such that its (conditional) coverage probability is close to the nominal. To do so, we pay attention to the effect of estimator variability such as asymptotic bias and mean square error. Numerical analysis is carried out to illustrate this calculation for the Autoregressive Conditional Heteroscedastic (ARCH) model, an observable volatility type model. In comparison, we find VaR for the latent volatility model i.e., the Stochastic Volatility Autoregressive (SVAR) model. It is found that the effect of estimator variability is significant to obtain VaR forecast with better coverage. In addition, we may only be able to assess unconditional coverage probability for VaR forecast of the SVAR model. This is due to the fact that the volatility process of the model is unobservable.

Backtesting Marginal Expected Shortfall and Related Systemic Risk Measures

This paper proposes an original approach for backtesting systemic risk measures. This backtesting approach makes it possible to assess the systemic risk measure forecasts used to identify the financial institutions that contribute the most to the overall risk in the financial system. Our procedure is based on simple tests similar to those generally used to backtest the standard market risk measures such as value-at-risk or expected shortfall. We introduce a concept of violation associated with the marginal expected shortfall (MES), and we define unconditional coverage and independence tests for these violations. We can generalize these tests to any MES-based systemic risk measures such as SES, SRISK, or ∆CoVaR. We study their asymptotic properties in the presence of estimation risk and investigate their finite sample performance via Monte Carlo simulations. An empirical application to a panel of U.S. financial institutions is conducted to assess the validity of MES, SRISK, and ∆CoVaR forecasts issued from a GARCH-DCC model. Our results show that this model provides valid forecasts for MES and SRISK when considering a medium-term horizon. Finally, we propose an early warning system indicator for future systemic crises deduced from these backtests. Our indicator quantifies how much is the measurement error issued by a systemic risk forecast at a given point in time which can serve for the early detection of global market reversals.

A VaR-based Downside Risk Analysis of Indian Equity Mutual Funds in the Pre- and Post-global Financial Crisis Periods

This article analyses downside risk of Indian equity mutual funds from 1999 to 2014 using a value at risk (VaR)-based approach. We use weekly return data of a sample of 349 equity mutual funds during the said period to estimate their weekly VaRs on a rolling basis using some parametric and non-parametric models. Moving average (MA), exponentially weighted MA and GARCH (1, 1) are the parametric models and historical simulation (HS) is the non-parametric model. We also carry out backtesting of the models using three popular approaches—two under the unconditional coverage approach, namely Jorion’s ‘Failure Rate’ approach and Kupiec’s proportion of ‘failures’ (POF) test, and one under the conditional coverage approach, namely the Christoffersen’s Independence test—to test the robustness of the VaR models. Our results show that Indian equity mutual funds exhibit considerable downside risk during the chosen period, in terms of the magnitude of the projected VaRs. Moreover, significant proportions of the funds ‘fail’ the predicted VaRs, particularly during times of crisis for some of the models, raising questions about their robustness in an investment setting in India. On the whole, both from failure proportion as well as backtesting perspective, the GARCH (1,1) seems to be the most robust of the models. JEL codes: G32, G15, G23

Real estate VaR estimation in Seoul and Busan, Korea

In this study, we estimtate the value-at-risk (VaR) of the 10-day average apartment prices of Gangnam-gu, Seoul and Haeundae-gu, Busan in Korea. For this purpose, we adopt the semiparametric quantile regression approach for the VaR calculation based on ARIMA-GARCH models, employing the well-known risk measures such as the conditional autoregressive value-at-risk (CAViaR) and conditional autoregressive expectile (CARE) methods. After conducting the unconditional coverage (UC) and conditional coverage (CC) tests on the estimated VaRs, we conclude that the two methods perform similarly for the Seoul case but the CARE method shows more stability than the CAViaR method in the Busan case.

Detecting underestimates of risk in VaR models

Abstract This paper explores avenues to improve VaR backtesting against violations of unconditional coverage, with emphasis on detecting underestimates of risk. I show that existing tests for unconditional coverage have an undesirable asymmetry in power, making them less (more) effective at detecting underestimates (overestimates) of risk. A set of Lagrange multiplier (LM) tests is proposed, leading to improved power against the more dangerous underestimates. Also, the use of one-sided coverage tests is revisited, with a one-sided version of the new LM test performing best. Results are supported by simulations, showing the effectiveness of the new tests in finite samples. The same issues of asymmetric power are shown to carry over to the case of conditional coverage, though it appears that finite sample issues affecting the independence component of the tests can adversely impact performance. A simple weighted test statistic is shown to help alleviate this problem, providing an avenue to exploit the advantages of the LM approach in a conditional coverage setting.

Backtesting expected shortfall: a simple recipe?

We propose a new backtesting framework for Expected Shortfall that could be used by the regulator. Instead of looking at the estimated capital reserve and the realised cash-flow separately, one could bind them into the secured position, for which risk measurement is much easier. Using this simple concept combined with monotonicity of Expected Shortfall with respect to its target confidence level we introduce a natural and efficient backtesting framework. Our test statistics is given by the biggest number of worst realisations for the secured position that add up to a negative total. Surprisingly, this simple quantity could be used to construct an efficient backtesting framework for unconditional coverage of Expected Shortfall in a natural extension of the regulatory traffic-light approach for Value-at-Risk. While being easy to calculate, the test statistic is based on the underlying duality between coherent risk measures and scale-invariant performance measures.

Forecasting Value at Risk: Evidence from Emerging Economies in Asia

In this paper, various Value-at-Risk techniques are applied to stock indices of 9 Asian emerging financial markets. The results from our selected models are then backtested by Unconditional Coverage, Independence, Joint Tests of Unconditional Coverage and Independence and Basel tests to ensure the quality of Value-at-Risk (VaR) estimates. The main conclusions are: (1) Timevarying volatility is the most important characteristic of stock returns when modelling VaR; (2) Financial data is not normally distributed, indicating that the normality assumption of VaR is not relevant; (3) Among VAR forecasting approaches, the backtesting based on in- and out-of-sample evaluations confirms its superiority in the class of GARCH models; Historical Simulation (HS), Filtered Historical Simulation (FHS), RiskMetrics and Monte Carlo were rejected because of its underestimation (for HS and RiskMetrics) or overestimation (for the FHS and Monte Carlo); (4) Models under student’s t and skew student’s t distribution are better in taking into account financial data’s characters; and (5) Forecasting VaR for futures index is harder than for stock index. Moreover, results show that there is no evidence to recommend the use of GARCH (1,1) to estimate VaR for all markets. In practice, the HS and RiskMetrics are popularly used by banks for large portfolios, despite of its serious underestimations of actual losses. These findings would be helpful for financial managers, investors and regulators dealing with stock markets in Asian emerging economies.

An Estimation of Value at Risk using GARCH Models for the Conventional and Islamic Stock Market in Malaysia

This study attempts to estimate the value-at-risk (VaR) to forecast volatility for both conventional and Islamic stock markets in Malaysia. In particular, the purpose of the article is to investigate whether GARCH models are accurate in the evaluation of VaR in emerging stock markets such as Malaysia.The daily return of the conventional (KLCI) and Islamic (FBMS) stock market are analysed for the period 2000 – 2015. The volatility model of GARCH (1,1), TGARCH (1,1) and CGARCH (1,1) with a normal and and student-t distribution are used to model the conditional variance of the stock market returns. The VaR violations of unconditional coverage and the backtesting procedure of Kupiec test are used to check the reliability and accuracy of the volatility model used for both normal and student-t distribution. Based on the Akaike Information Criterion (AIC), the best model for modelling the conventional and Islamic stock market returns is TGARCH (1,1). The backtesting results showed that for all GARCH models used, the normal distribution gives better forecast VaR compared to the student’s t distribution.

Bitcoin returns and risk: A general GARCH and GAS analysis

This paper performs a general GARCH and GAS analysis for modelling and forecasting bitcoin returns and risk. Since Bitcoin trading exhibits excess volatility compared with other securities, it is important to model its risk and returns. We consider heavy-tailed GARCH models as well as GAS models based on the score function of the predictive conditional density of the bitcoin returns. We compare out-of-sample 1%-Value-at-Risk (VaR) forecasts under 45 different specifications using three backtesting procedures. We find that GAS models with heavy-tailed distributions provide the best out-of-sample forecast and goodness-of-fit properties to bitcoin returns and risk modelling. Normally-distributed GARCH models are always outperformed by heavy-tailed GARCH or GAS models. Besides, heavy-tailed GAS models provide the best conditional and unconditional coverage for 1%-VaR forecasts, illustrating the importance of modelling excess kurtosis for bitcoin returns. Hence, our findings have important implications for risk managers and investors for using bitcoin in optimal hedging or investment strategies.

Spectral Backtests of Forecast Distributions with Application to Risk Management

We study a class of backtests for forecast distributions in which the test statistic is a spectral transformation that weights exceedance events by a function of the modeled probability level. The choice of the kernel function makes explicit the user's priorities for model performance. The class of spectral backtests includes tests of unconditional coverage and tests of conditional coverage. We show how the class embeds a wide variety of backtests in the existing literature, and propose novel variants as well. In an empirical application, we backtest forecast distributions for the overnight P&L of ten bank trading portfolios. For some portfolios, test results depend materially on the choice of kernel.

Bias corrected empirical Bayes confidence intervals for the selected mean

ABSTRACTEmpirical Bayes (EB) methods are very useful for post selection inference. Following Datta et al. (2002), we construct EB confidence intervals for the selected population mean. The EB intervals are adjusted to achieve the target coverage probabilities asymptotically up to the second order. Both unconditional coverage probabilities of EB intervals and corresponding probabilities conditional on ancillary statistics are found.

Forecasting Value-at-Risk (VaR) in the Major Asian Economies

This is an empirical study of forecasting Value-at-Risk (VaR) in the major Asian economies. The VaR is first forecasted for Singapore, Malaysia, Hong Kong of China, Indonesia, South Korea, Philippines, Thailand, China, Taiwan of China and India using different competing models. The VaR estimates are then backtested using unconditional coverage test, conditional coverage test and loss function to arrive at the best VaR model for each of the economies. The results are mixed with the highest success rate of FIGARCH model. Also, the appropriateness of the models changes across quantiles and between tails.

New backtests for unconditional coverage of expected shortfall

While value-at-risk has been the standard risk measure for a long time, expected shortfall (ES) has become more and more popular in recent times, as it provides important information about tail risk. We present a new backtest for the unconditional coverage property of the ES. The test is based on the so-called cumulative violation process, and its main advantage is that the distribution is known for finite out-of-sample size. This leads to better size and power properties compared with existing tests. Moreover, we extend the test principle to a multivariate test and analyze its behavior via simulations and an application to bank returns.