Bayesian realized-GARCH models for financial tail risk forecasting incorporating the two-sided Weibull distribution

The realized-GARCH framework is extended to incorporate the two-sided Weibull distribution, for the purpose of volatility and tail risk forecasting in a financial time series. Further, the realized range, as a competitor for realized variance or daily returns, is employed as the realized measure in the realized-GARCH framework. Sub-sampling and scaling methods are applied to both the realized range and realized variance, to help deal with inherent micro-structure noise and inefficiency. A Bayesian Markov Chain Monte Carlo (MCMC) method is adapted and employed for estimation and forecasting, while various MCMC efficiency and convergence measures are employed to assess the validity of the method. In addition, the properties of the MCMC estimator are assessed and compared with maximum likelihood, via a simulation study. Compared to a range of well-known parametric GARCH and realized-GARCH models, tail risk forecasting results across seven market indices, as well as two individual assets, clearly favour the proposed realized-GARCH model incorporating the two-sided Weibull distribution; especially those employing the sub-sampled realized variance and sub-sampled realized range.     

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.     

Semi-parametric Bayesian tail risk forecasting incorporating realized measures of volatility

Realized measures employing intra-day sources of data have proven effective for dynamic volatility and tail-risk estimation and forecasting. Expected shortfall (ES) is a tail risk measure, now recommended by the Basel Committee, involving a conditional expectation that can be semi-parametrically estimated via an asymmetric sum of squares function. The conditional autoregressive expectile class of model, used to implicitly model ES, has been extended to allow the intra-day range, not just the daily return, as an input. This model class is here further extended to incorporate information on realized measures of volatility, including realized variance and realized range (RR), as well as scaled and smoothed versions of these. An asymmetric Gaussian density error formulation allows a likelihood that leads to direct estimation and one-step-ahead forecasts of quantiles and expectiles, and subsequently of ES. A Bayesian adaptive Markov chain Monte Carlo method is developed and employed for estimation and forecasting. In an empirical study forecasting daily tail risk measures in six financial market return series, over a seven-year period, models employing the RR generate the most accurate tail risk forecasts, compared to models employing other realized measures as well as to a range of well-known competitors.     

Model risk of contingent claims

Paralleling regulatory developments, we devise value-at-risk and expected shortfall type risk measures for the potential losses arising from using misspecified models when pricing and hedging contingent claims. Essentially, P&L from model risk corresponds to P&L realized on a perfectly hedged position. Model uncertainty is expressed by a set of pricing models, each of which represents alternative asset price dynamics to the model used for pricing. P&L from model risk is determined relative to each of these models. Using market data, a unified loss distribution is attained by weighing models according to a likelihood criterion involving both calibration quality and model parsimony. Examples demonstrate the magnitude of model risk and corresponding capital buffers necessary to sufficiently protect trading book positions against unexpected losses from model risk. A further application of the model risk framework demonstrates the calculation of gap risk of a barrier option when employing a semi-static hedging strategy.     

On risk management problems related to a coherence property

Value at Risk has lost the battle against Expected Shortfall on theoretical grounds, the latter satisfying all coherence properties while the former may, on carefully constructed cases, lack the sub-additivity property that is in a sense, the most important property a risk measure ought to satisfy. While the superiority of Expected Shortfall is evident as a theoretical tool, little has been researched on the properties of estimators proposed in the literature. Since those estimators are the real tools for calculating bank capital reserves in practice, the natural question that one may ask is whether a given estimator of Expected Shortfall also satisfies the coherence properties. In this paper, we show that it is possible to have estimators of Expected Shortfall that do not satisfy the sub-additivity condition. This finding should motivate risk managers and quantitative asset managers to investigate further the properties of the estimators of the risk measures they are currently utilizing.     

Model risk of risk models

This paper evaluates the model risk of models used for forecasting systemic and market risk. Model risk, which is the potential for different models to provide inconsistent outcomes, is shown to be increasing with market uncertainty. During calm periods, the underlying risk forecast models produce similar risk readings; hence, model risk is typically negligible. However, the disagreement between the various candidate models increases significantly during market distress, further frustrating the reliability of risk readings. Finally, particular conclusions on the underlying reasons for the high model risk and the implications for practitioners and policy makers are discussed.     

Fast simulations in credit risk

We consider the problem of simulating tail loss probabilities and expected losses conditioned on exceeding a large threshold (expected shortfall) for credit portfolios. Our new idea, called the geometric shortcut, allows an efficient simulation for the case of independent obligors. It is even possible to show that, when the average default probability tends to zero, its asymptotic efficiency is higher than that of the naive algorithm. The geometric shortcut is also useful for models with dependent obligors and can be used for dependence structures modeled with arbitrary copulae. The paper contains the details for simulating the risk of the normal copula credit risk model by combining outer importance sampling with the geometric shortcut. Numerical results show that the new method is efficient in assessing tail loss probabilities and expected shortfall for credit risk portfolios. The new method outperforms all known methods, especially for credit portfolios consisting of weakly correlated obligors and for evaluating the tail loss probabilities at many thresholds in a single simulation run.     

Which are the SIFIs? A Component Expected Shortfall approach to systemic risk

This paper proposes a component approach to systemic risk which allows to decompose the risk of the aggregate financial system (measured by Expected Shortfall) while accounting for the firm characteristics. Developed by analogy with the Component Value-at-Risk concept, our new systemic risk measure, called Component ES, presents several advantages. It is a hybrid measure, which combines the Too Interconnected To Fail and the Too Big To Fail logics. CES relies only on publicly available daily data and encompasses the popular Marginal ES measure. CES can be used to assess the contribution of a firm to systemic risk at a precise date but also to forecast its contribution over a certain period. The empirical application verifies the ability of CES to identify the most systemically risky firms during the 2007–2009 financial crisis. We show that our measure identifies the institutions labeled as SIFIs by the Financial Stability Board.     

European systemic credit risk transmission using Bayesian networks

The analysis of systemic credit risk is one of the most important concerns within the financial system. Its complexity lies in adequately measuring how the transmission of systemic default spreads through assets or financial markets. The transmission structure of systemic credit risk across several European sectoral CDS is studied by dynamic Bayesian networks. The new approach allows for a more advanced analysis of systemic risk transmission, including long-term and more complex relationships. The modelling reveals as relevant only relationships between the original series and one- and two-lagged series. Network structure learning displays a robust and stationary underlying risk transmission structure, pointing to a consolidated transmission mechanism of systemic credit risk between CDSs. Between 5 % and 40 % of sectoral CDS series variances are explained by the network relationships. The modelling allows us to ascertain which relationships between the CDS series show positive (amplifier) and negative (reducer) effects of systemic risk transmission.     

Estimating portfolio risk for tail risk protection strategies

We forecast portfolio risk for managing dynamic tail risk protection strategies, based on extreme value theory, expectile regression, copula‐GARCH and dynamic generalized autoregressive score models. Utilizing a loss function that overcomes the lack of elicitability for expected shortfall, we propose a novel expected shortfall (and value‐at‐risk) forecast combination approach, which dominates simple and sophisticated standalone models as well as a simple average combination approach in modeling the tail of the portfolio return distribution. While the associated dynamic risk targeting or portfolio insurance strategies provide effective downside protection, the latter strategies suffer less from inferior risk forecasts, given the defensive portfolio insurance mechanics.     

A practical approach for the evaluation of acceptable risk in road tunnels

In many European countries it is common to adopt quantitative criteria in evaluation of acceptable risk in road tunnels. Such criteria, usually expressed by FN-criteria and IR-values, will easily lead to a regime that is difficult to adopt in practice, as the use of such criteria requires extensive analyses and documentation for all types of tunnels. In this paper, a more practical approach for the evaluation of acceptable risk in road tunnels is presented, in which quantitative risk acceptance criteria are used for some road tunnels, while qualitative criteria are used for others. This means that varying degrees of effort and documentation are necessary for the evaluation of acceptable risk in road tunnels. The approach suggested is inspired by challenges in Norway.     

A risk assessment model for banks

Summary. This paper proposes a model to assess risk for banks. Its main innovation is to incorporate endogenous interaction among banks, where the actual risk an individual bank bears also depends on its interaction with other banks and investors. We develop a two-period general equilibrium model with three active heterogeneous banks, incomplete markets, and endogenous default. The model is calibrated against UK banking data and therefore can be implemented as a risk assessment tool for regulators and central banks. We address the impact of monetary and regulatory policy, credit and capital shocks in the real and financial sectors.We are grateful to Lea Zicchino, an anonymous referee, seminar participants at the Bank of England, and the third Conference in Research in Economic Theory and Econometrics, Syros, for helpful comments. The views expressed are those of the authors and do not necessarily reflect those of the Bank of England.This revised version was published online in January 2005 with corrections to the Cover date.     

A comprehensive approach to measuring the relation between systemic risk exposure and sovereign debt

Using an integrated model to control for simultaneity, as well as new risk measurement techniques such as Adapted Exposure CoVaR and Marginal Expected Shortfall (MES), we show that the aggregate systemic risk exposure of financial institutions is positively related to sovereign debt yields in European countries in an episodic manner, varying positively with the intensity of the financial crisis facing a particular nation. We find evidence of a simultaneous relation between systemic risk exposure and sovereign debt yields. This suggests that models of sovereign debt yields should also include the systemic risk of a country's financial system in order to avoid potentially important mis-specification errors. We find evidence that systemic risk of a country's financial institutions and the risk of sovereign governments are inter-related and shocks to these domestic linkages are stronger and longer lasting than international risk spillovers. Thus, the channel in which domestic sovereign debt yields can be affected by another nation's sovereign debt is mostly an indirect one in that shocks to a foreign country's government finances are transmitted to that country's financial system which, in turn, can spill over to the domestic financial system and, ultimately, have a destabilizing effect on the domestic sovereign debt market.     

An analysis of a three-factor model proposed by the Danish Society of Actuaries for forecasting and risk analysis

This paper provides the explicit solution to the three-factor diffusion model recently proposed by the Danish Society of Actuaries to the Danish industry of life insurance and pensions. The solution is obtained by use of the known general solution to multidimensional linear stochastic differential equation systems. With offset in the explicit solution, we establish the conditional distribution of the future state variables which allows for exact simulation. Using exact simulation, we illustrate how simulation of the system can be improved compared to a standard Euler scheme. In order to analyze the effect of choosing the exact simulation scheme over the traditional Euler approximation scheme frequently applied by practitioners, we carry out a simulation study. We show that due to its recursive nature, the Euler scheme becomes computationally expensive as it requires a small step size in order to minimize discretization errors. Using our exact simulation scheme, one is able to cut these computational costs significantly and obtain even better forecasts. As probability density tail behavior is key to expected investment portfolio performance, we further conduct a risk analysis in which we compare well-known risk measures under both schemes. Finally, we conduct a sensitivity analysis and find that the relative performance of the two schemes depends on the chosen model parameter estimates.     

Time-varying beta risk of Pan-European industry portfolios: A comparison of alternative modeling techniques

This paper investigates the time-varying behavior of systematic risk for 18 pan-European sectors. Using weekly data over the period 1987–2005, six different modeling techniques in addition to the standard constant coefficient model are employed: a bivariate t-GARCH(1,1) model, two Kalman filter (KF)-based approaches, a bivariate stochastic volatility model estimated via the efficient Monte Carlo likelihood technique as well as two Markov switching models. A comparison of ex-ante forecast performances of the different models indicate that the random walk process in connection with the KF is the preferred model to describe and forecast the time-varying behavior of sector betas in a European context.     

Modeling the business risk of financially weakened firms: A new approach for corporate bond pricing

Most structural models of default risk assume that the firm's asset return is normally distributed, with a constant volatility. By contrast, this article details the properties that the process of assets should have in the case of financially weakened firms. It points out that jump-diffusion processes with time-varying volatility provide a refined and accurate perspective on the business risk dimension of default risk. Representative Arrow-Debreu state price densities (SPD) and term structures of credit spreads are then explored. The credit curves show that the business uncertainties play a major in the pricing of corporate liabilities.     

A conceptual Object-Oriented Bayesian Network (OOBN) for modeling aircraft carrier-based UAS safety risk

This paper illustrates the conceptual development of a demonstration Object-Oriented Bayesian Network (OOBN) to integrate the hazards associated with an experimental Unmanned Aircraft System (UAS) planned for deployment from an aircraft carrier. The final Air/Ship Integration (A/SI) demonstration model is characterized by a top-level Bayesian network model with nine sub-nets comprising 70 causal factors with 15 mitigations. With the creation of a probabilistic model, inferences about changes to the states of the causal factors given the presence or absence of controls or mitigations can be ascertained. These inferences build on qualitative reasoning and enable an analyst to identify the most prominent causal factor groupings leading to a prioritization of the most influential causal factors. Mitigation effects can be systematically studied and assessed. The A/SI OOBN demonstration model illustrates the construction of an integrative safety risk model that may be used to compute a higher-order system mishap probability for an experimental UAS that interacts with ship operations in a highly severe, dynamic sea environment. In addition to computing mishap probabilities, the Bayesian approach may also be used to support control contingency management for possible mitigation implementation.     

A new approach to measuring riskiness in the equity market: Implications for the risk premium

We introduce a new approach to measuring riskiness in the equity market. We propose option implied and physical measures of riskiness and investigate their performance in predicting future market returns. The predictive regressions indicate a positive and significant relation between time-varying riskiness and expected market returns. The significantly positive link between aggregate riskiness and market risk premium remains intact after controlling for the S&P 500 index option implied volatility (VIX), aggregate idiosyncratic volatility, and a large set of macroeconomic variables. We also provide alternative explanations for the positive relation by showing that aggregate riskiness is higher during economic downturns characterized by high aggregate risk aversion and high expected returns.     

A new technique for calibrating stochastic volatility models: the Malliavin gradient method

We discuss the application of gradient methods to calibrate mean reverting stochastic volatility models. For this we use formulas based on Girsanov transformations as well as a modification of the Bismut–Elworthy formula to compute the derivatives of certain option prices with respect to the parameters of the model by applying Monte Carlo methods. The article presents an extension of the ideas to apply Malliavin calculus methods in the computation of Greek's.