Pitfalls in backtesting Historical Simulation VaR models

Historical Simulation (HS) and its variant, the Filtered Historical Simulation (FHS), are the most popular Value-at-Risk forecast methods at commercial banks. These forecast methods are traditionally evaluated by means of the unconditional backtest. This paper formally shows that the unconditional backtest is always inconsistent for backtesting HS and FHS models, with a power function that can be even smaller than the nominal level in large samples. Our findings have fundamental implications in the determination of market risk capital requirements, and also explain Monte Carlo and empirical findings in previous studies. We also propose a data-driven weighted backtest with good power properties to evaluate HS and FHS forecasts. A Monte Carlo study and an empirical application with three US stocks confirm our theoretical findings. The empirical application shows that multiplication factors computed under the current regulatory framework are downward biased, as they inherit the inconsistency of the unconditional backtest.     

Backtesting for risk-based regulatory capital

In this paper we present a framework for backtesting all currently popular risk measurement methods for quantifying market risk (including value-at-risk and expected shortfall) using the functional delta method. Estimation risk can be taken explicitly into account. Based on a simulation study we provide evidence that tests for expected shortfall with acceptable low levels have a better performance than tests for value-at-risk in realistic financial sample sizes. We propose a way to determine multiplication factors, and find that the resulting regulatory capital scheme using expected shortfall compares favorably to the current Basel Accord backtesting scheme.     

Integrating market and credit risk: A simulation and optimisation perspective

We introduce a modelling paradigm which integrates credit risk and market risk in describing the random dynamical behaviour of the underlying fixed income assets. We then consider an asset and liability management (ALM) problem and develop a multistage stochastic programming model which focuses on optimum risk decisions. These models exploit the dynamical multiperiod structure of credit risk and provide insight into the corrective recourse decisions whereby issues such as the timing risk of default is appropriately taken into consideration. We also present an index tracking model in which risk is measured (and optimised) by the CVaR of the tracking portfolio in relation to the index. In-sample as well as out-of-sample (backtesting) experiments are undertaken to validate our approach. The main benefits of backtesting, that is, ex-post analysis are that (a) we gain insight into asset allocation decisions, and (b) we are able to demonstrate the feasibility and flexibility of the chosen framework.     

Risk quantification for commodity ETFs: Backtesting value-at-risk and expected shortfall

This paper calibrates risk assessment of alternative methods for modeling commodity ETFs. We implement recently proposed backtesting techniques for both value-at-risk (VaR) and expected shortfall (ES) under parametric and semi-nonparametric techniques. Our results indicate that skewed-t and Gram-Charlier distributional assumptions present the best relative performance for individual Commodity ETFs for those confidence levels recommended by Basel Accords. In view of these results, we recommend the application of leptokurtic distributions and semi-nonparametric techniques to mitigate regulation concerns about global financial stability of commodity business.     

Scaling models for the severity and frequency of external operational loss data

According to Basel II criteria, the use of external data is indispensable to the implementation of an advanced method for calculating operational risk capital. This article investigates how the severity and frequencies of external losses are scaled for integration with internal data. We set up an initial model designed to explain the loss severity by taking into account potential selection bias in the external data. Estimation results show that many variables have significant power in explaining the loss amount. We use them to develop a normalization formula. We develop a zero-inflated count-data model to scale the loss frequency. We compute an operational VaR and we conduct out-of-sample backtesting.     

An analysis of VaR-based capital requirements

We study the behavior of a financial institution subject to capital requirements based on self-reported VaR measures, as in the Basel Committee's Internal Models Approach. We view these capital requirements and the associated backtesting procedure as a mechanism designed to induce financial institutions to reveal the risk of their investments and to support this risk with adequate levels of capital. Accordingly, we consider the simultaneous choice of an optimal dynamic reporting and investment strategy. Overall, we find that VaR-based capital requirements can be very effective not only in curbing portfolio risk but also in inducing revelation of this risk.     

Estimation of multiple period expected shortfall and median shortfall for risk management

With the regulatory requirements for risk management, Value at Risk (VaR) has become an essential tool in determining capital reserves to protect the risk induced by adverse market movements. The fact that VaR is not coherent has motivated the industry to explore alternative risk measures such as expected shortfall. The first objective of this paper is to propose statistical methods for estimating multiple-period expected shortfall under GARCH models. In addition to the expected shortfall, we investigate a new tool called median shortfall to measure risk. The second objective of this paper is to develop backtesting methods for assessing the performance of expected shortfall and median shortfall estimators from statistical and financial perspectives. By applying our expected shortfall estimators and other existing approaches to seven international markets, we demonstrate the superiority of our methods with respect to statistical and practical evaluations. Our expected shortfall estimators likely provide an unbiased reference for setting the minimum capital required for safeguarding against expected loss.     

Effectiveness of copula-extreme value theory in estimating value-at-risk: empirical evidence from Asian emerging markets

A traditional Monte Carlo simulation using linear correlations induces estimation bias in measuring portfolio value-at-risk (VaR), due to the well-documented existence of fat-tail, skewness, truncations, and non-linear relations in return distributions. In this paper, we consider the above issues in modeling VaR and evaluate the effectiveness of using copula-extreme-value-based semiparametric approaches. To assess portfolio risk in six Asian markets, we incorporate a combination of extreme value theory (EVT) and various copulas to build joint distributions of returns. A backtesting analysis using a Monte Carlo VaR simulation suggests that the Clayton copula-EVT evinces the best performance regardless of the shapes of the return distributions, and that in general the copulas with the EVT provide better estimations of VaRs than the copulas with conventionally employed empirical distributions. These findings still hold in conditional-coverage-based backtesting. These findings indicate the economic significance of incorporating the down-side shock in risk management.     

An empirical investigation of multiperiod tail risk forecasting models

In the context of multiperiod tail risk (i.e., VaR and ES) forecasting, we provide a new semiparametric risk model constructed based on the forward-looking return moments estimated by the stochastic volatility model with price jumps and the Cornish–Fisher expansion method, denoted by SVJCF. We apply the proposed SVJCF model to make multiperiod ahead tail risk forecasts over multiple forecast horizons for S&P 500 index, individual stocks and other representative financial instruments. The model performance of SVJCF is compared with other classical multiperiod risk forecasting models via various backtesting methods. The empirical results suggest that SVJCF is a valid alternative multiperiod tail risk measurement; in addition, the tail risk generated by the SVJCF model is more stable and thus should be favored by risk managers and regulatory authorities.     

Tail risk of coal futures in China's market

This study evaluates the downside tail risk of coal futures contracts (coke, coking coal and thermal coal) traded in the Chinese market between 2011 and 2021, measured by value at risk (VaR). We examine the one-day-ahead VaR forecasting performance with a hybrid econometric and deep learning model (GARCH-LSTM), GARCH family models, extreme value theory models, quantile regression models and two naïve models (historical simulation and exponentially weighted moving average). We use four backtesting techniques and the model confidence set to identify the optimal models. The results suggest that the models focusing on tail risk or utilising long short-term memory generate more effective risk management.     

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk (VaR) and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.     

Does prudential regulation contribute to effective measurement and management of interest rate risk? Evidence from Italian banks

This paper contributes to prior literature and to the current debate concerning recent revisions of the regulatory approach to measuring bank exposure to interest rate risk in the banking book by focusing on assessment of the appropriate amount of capital banks should set aside against this specific risk. We first discuss how banks might develop internal measurement systems to model changes in interest rates and measure their exposure to interest rate risk that are more refined and effective than are regulatory methodologies. We then develop a backtesting framework to test the consistency of methodology results with actual bank risk exposure. Using a representative sample of Italian banks between 2006 and 2013, our empirical analysis supports the need to improve the standardized shock currently enforced by the Basel Committee on Banking Supervision. It also provides useful insights for properly measuring the amount of capital to cover interest rate risk that is sufficient to ensure both financial system functioning and banking stability.     

Risk models-at-risk

The experience from the global financial crisis has raised serious concerns about the accuracy of standard risk measures as tools for the quantification of extreme downward risks. A key reason for this is that risk measures are subject to a model risk due, e.g. to specification and estimation uncertainty. While regulators have proposed that financial institutions assess the model risk, there is no accepted approach for computing such a risk. We propose a remedy for this by a general framework for the computation of risk measures robust to model risk by empirically adjusting the imperfect risk forecasts by outcomes from backtesting frameworks, considering the desirable quality of VaR models such as the frequency, independence and magnitude of violations. We also provide a fair comparison between the main risk models using the same metric that corresponds to model risk required corrections.     

Estimating loss-given default through advanced credibility theory

The New Basel Accord allows internationally active banking organizations to calculate their credit risk capital requirements using an internal ratings based approach, subject to supervisory review. One of the modeling components is the loss-given default (LGD): it represents the credit loss for a bank when extreme events occur that influence the obligor ability to repay his debts to the bank. Among researchers and practitioners the use of statistical models such as linear regression, Tobit or decision trees is quite common in order to compute LGDs as a forecasting of historical losses. However, these statistical techniques do not seem to provide robust estimation and show low performance. These results could be driven by some factors that make differences in LGD, such as the presence and quality of collateral, timing of the business cycle, workout process management and M&A activity among banks. This paper evaluates an alternative method of modeling LGD using a technique based on advanced credibility theory typically used in actuarial modeling. This technique provides a statistical component to the credit and workout experts’ opinion embedded in the collateral and workout management process and improve the predictive power of forecasting. The model has been applied to an Italian Bank Retail portfolio represented by Overdrafts; the application of credibility theory provides a higher predictive power of LGD estimation and an out-of-time sample backtesting has shown a stable accuracy of estimates with respect to the traditional LGD model.     

Tracking bond indices in an integrated market and credit risk environment

Abstract

The management of credit risky assets requires simulation models that integrate the disparate sources of credit and market risk, and suitable optimization models for scenario analysis. In this paper we integrate Monte Carlo simulation models for credit risk with scenario optimization, and develop a methodology for tracking broadly defined corporate bond indices. Testing of the models shows that the integration of the multiple risk factors improves significantly the performance of tracking models. Good tracking performance can be achieved by optimizing strategic asset allocation among broad classes of corporate bonds. However, extra value is generated with a tactical model that optimizes bond picking decisions as well. It is also shown that adding small corporate bond holdings in portfolios that track government bond indices improves the risk/return characteristics of the portfolios. The empirical results to substantiate the findings of this study are obtained by backtesting the model over a recent 30 month period.     

Portfolio optimization for heavy-tailed assets: Extreme Risk Index vs. Markowitz

Using daily returns of the S&P 500 stocks from 2001 to 2011, we perform a backtesting study of the portfolio optimization strategy based on the Extreme Risk Index (ERI). This method uses multivariate extreme value theory to minimize the probability of large portfolio losses. With more than 400 stocks to choose from, our study seems to be the first application of extreme value techniques in portfolio management on a large scale. The primary aim of our investigation is the potential of ERI in practice. The performance of this strategy is benchmarked against the minimum variance portfolio and the equally weighted portfolio. These fundamental strategies are important benchmarks for large-scale applications. Our comparison includes annualized portfolio returns, maximal drawdowns, transaction costs, portfolio concentration, and asset diversity in the portfolio. In addition to that we study the impact of an alternative tail index estimator. Our results show that the ERI strategy significantly outperforms both the minimum-variance portfolio and the equally weighted portfolio on assets with heavy tails.     

The implication of banking competition: Evidence from African countries

This paper examines how bank efficiency and stability are affected by the market power in Africa. Our results show that the higher degree of market power is associated with high level of efficiency and profitability. The banks with more market power operating are able to be in command of the price and hence improve their profit. The market power has a benefit in both stability and risk. Hence, gain in market will increase the stability and reduce the risk for banking system. Our findings do not support the argument that competition should not be based on a “quiet life hypothesis”.     

Predicting VaR for China's stock market: A score-driven model based on normal inverse Gaussian distribution

Under the framework of dynamic conditional score, we propose a parametric forecasting model for Value-at-Risk based on the normal inverse Gaussian distribution (Hereinafter NIG-DCS-VaR), which creatively incorporates intraday information into daily VaR forecast. NIG specifies an appropriate distribution to return and the semi-additivity of the NIG parameters makes it feasible to improve the estimation of daily return in light of intraday return, and thus the VaR can be explicitly obtained by calculating the quantile of the re-estimated distribution of daily return. We conducted an empirical analysis using two main indexes of the Chinese stock market, and a variety of backtesting approaches as well as the model confidence set approach prove that the VaR forecasts of NIG-DCS model generally gain an advantage over those of realized GARCH (RGARCH) models. Especially when the risk level is relatively high, NIG-DCS-VaR beats RGARCH-VaR in terms of coverage ability and independence.     

Score-driven copula models for portfolios of two risky assets

ABSTRACT

The precise measurement of the association between asset returns is important for financial investors and risk managers. In this paper, we focus on a recent class of association models: Dynamic Conditional Score (DCS) copula models. Our contributions are the following: (i) We compare the statistical performance of several DCS copulas for several portfolios. We study the Clayton, rotated Clayton, Frank, Gaussian, Gumbel, rotated Gumbel, Plackett and Student's t copulas. We find that the DCS model with the Student's t copula is the most parsimonious model. (ii) We demonstrate that the copula score function discounts extreme observations. (iii) We jointly estimate the marginal distributions and the copula, by using the Maximum Likelihood method. We use DCS models for mean, volatility and association of asset returns. (iv) We estimate robust DCS copula models, for which the probability of a zero return observation is not necessarily zero. (v) We compare different patterns of association in different regions of the distribution for different DCS copulas, by using density contour plots and Monte Carlo (MC) experiments. (vi) We undertake a portfolio performance study with the estimation and backtesting of MC Value-at-Risk for the DCS model with the Student's t copula.     

Stock price crash risk and CEO power: Firm-level analysis

This study examines the impact of stock price crash risk on future CEO power. Using a large panel sample with 17,816 firm-year observations, we posit and find a significant negative impact of stock price crash risk on CEO power, suggesting that CEO power becomes smaller after stock price crashes. We also find that our results are stronger for firms with female CEOs and are largely driven by firms with shorter-tenure CEOs. In addition, we find that the significant negative impact of stock price crash risk on CEO power is diminished for firms with strong corporate governance. Our study responds to the call in Habib, Hasan, and Jiang (2018) by providing more empirical evidence on the consequences of stock price crash risk.