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Stress Testing in a Value at Risk Framework
Abstract
This article proposes a methodology-that can be used to parameterize stress test scenarios using the conditional probability distributions that are commonly used in daily VaR calculations. This new approach allows for a complete characterization of the value change distribution of a porfolio in a stress scenario. Statistical evidence demonstrates that the proposed loss exposure measure is substantially more accurate than the stress exposure measures that financial institutions commonly use.The results also suggest, contrary to popular perception, that historical VaR risk factor covari-ances and the assumption of conditional normality can be used to construct reasonably accurate loss exposure estimates in many stressful market environments.
... In our numerical verification, the correlation matrices to be approximated are calculated from the sample data in the Shenzhen Stock Exchange and the Shanghai Stock Exchange in China. For the constraints of (6.1), we notice that particular restrictions may be associated with a historical stressful event (such as the 1987 stock market crash and 2008 economic crisis), or can be a set of hypothetical changes related with same possible future stressful market event [13]. Generally, identifying accurately the stress events set and restricting the correlation coefficients between stress events and other underlying events [13] are very difficult, and Content courtesy of Springer Nature, terms of use apply. ...
... For the constraints of (6.1), we notice that particular restrictions may be associated with a historical stressful event (such as the 1987 stock market crash and 2008 economic crisis), or can be a set of hypothetical changes related with same possible future stressful market event [13]. Generally, identifying accurately the stress events set and restricting the correlation coefficients between stress events and other underlying events [13] are very difficult, and Content courtesy of Springer Nature, terms of use apply. Rights reserved. ...
In this paper, we are concerned with efficient algorithms for solving the least squares semidefinite programming which contains many equalities and inequalities constraints. Our proposed method is built upon its dual formulation and is a type of active-set approach. In particular, by exploiting the nonnegative constraints in the dual form, our method first uses the information from the Barzlai–Borwein step to estimate the active/inactive sets, and within an adaptive framework, it then accelerates the convergence by switching the L-BFGS iteration and the semi-smooth Newton iteration dynamically. We show the global convergence under mild conditions, and furthermore, the local quadratic convergence under the additional nondegeneracy condition. Various types of synthetic as well as real-world examples are tested, and preliminary but promising numerical experiments are reported.
... For this purpose, practitioners in financial industry usually seek an approximation of the restricted Table 1 Numerical results on E1 In our experiment, the correlation matrices to be approximated are calculated from the sample data in the Shenzhen Stock Exchange and the Shanghai Stock Exchange. We remark that the specified restrictions may correspond to an historical stressful event (such as the 1987 stock market crash and 2008 economic crisis) or may be a set of hypothetical changes corresponding to some possible future stressful market event (Kupiec 1998;Han et al. 2017; . But, anyhow, identifying the stress events set and restricting the correlation coefficients between stress events and other underlying events is a rather complicated and professional job (Kupiec 1998). ...
... We remark that the specified restrictions may correspond to an historical stressful event (such as the 1987 stock market crash and 2008 economic crisis) or may be a set of hypothetical changes corresponding to some possible future stressful market event (Kupiec 1998;Han et al. 2017; . But, anyhow, identifying the stress events set and restricting the correlation coefficients between stress events and other underlying events is a rather complicated and professional job (Kupiec 1998). For simplicity, we therefore randomly generate the constraints and their positions to imitate the stress testing scenarios. ...
This paper proposes an L-BFGS algorithm based on the active set technique to solve the matrix approximation problem: given a symmetric matrix, find a nearest approximation matrix in the sense of Frobenius norm to make it satisfy some linear equalities, inequalities and a positive semidefinite constraint. The problem is a convex optimization problem whose dual problem is a nonlinear convex optimization problem with non-negative constraints. Under the Slater constraint qualification, it has zero duality gap with the dual problem. To handle large-scale dual problem, we make use of the active set technique to estimate the active constraints, and then the L-BFGS method is used to accelerate free variables. The global convergence of the proposed algorithm is established under certain conditions. Finally, we conduct some preliminary numerical experiments, and compare the L-BFGS method with the inexact smoothing Newton method, the projected BFGS method, the alternating direction method and the two-metric projection method based on the L-BFGS. The numerical results show that our algorithm has some advantages in terms of CPU time when a large number of linear constraints are involved.
... Nor has the private sector gone further in formulating a palpable definition of stresstesting. Anecdotal evidence suggests that risk managers pursue stress-testing as a way of understanding gamma-risk and as a tool for studying portfolio allocation (Kupiec (1999)). Large and complex portfolios containing assets with nonlinear payoffs, such as options, may behave very differently in response to large shocks than would be expected given its valuation in more typical situations. ...
... factors to include in the scenario are basically a judgement call. Kupiec (1999) notes that when constructing a particular scenario, it is common practice to "zero out" all but the primary factors of interest. For example, any exchange rates that are not expected to play a key role in a middle east crisis scenario would be left unchanged --unlike the basic running of the risk model in which all factors move. ...
In recent months and years both practitioners and regulators have embraced the ideal of supplementing VaR estimates with "stress-testing". Risk managers are beginning to place an emphasis and expend resources on developing more and better stress-tests. In the present paper, we hold the standard approach to stress-testing up to a critical light. The current practice is to stress-test outside the basic risk model. Such an approach yields two sets of forecasts -- one from the stress-tests and one from the basic model. The stress scenarios, conducted outside the model, are never explicitly assigned probabilities. As such, there is no guidance as to the importance or revelance of the results of stress-tests. Moreover, how to combine the two forecasts into a usable risk metric is not known. Instead, we suggest folding the stress-tests into the risk model, thereby requiring all scenarios to be assigned probabilities.
... The paper makes three contributions. First, portfolio credit loss distributions are estimated using a non-parametric, stress-scenario approach (see Kupiec 1998 If loan sizes and LGDs were fixed instead of random, the estimated portfolio loss distributions would approximate transformed binomial distributions in which an aggregate annual borrower default rate is the key parameter, so I describe this paper's approach as involving modified binomial loss distributions. Because aggregate default rates can be related to the severity of economic downturns and (in this paper's setup) loss distribution percentiles represent bank survival rates, policymakers may set capital to limit bank failures to some acceptable estimated rate in an economic scenario of intuitively specified severity. ...
... Of course, in an even worse recession, the bank failure rate would be higher. This paper's way of defining and modeling VaR loss distributions is related to existing stresstest methods of capital allocation (see Jorion 2001, Kupiec 1998, and Shepheard-Walwyn and Rohner 2000. However, a typical credit stress-test analysis specifies default rates for each line of business, or for firms in each geographic region or industry. ...
Resampling implementation of a stress-scenario approach to estimating portfolio default loss distributions is proposed as the basis for estimates of the appropriate absolute level of economic capital allocations for portfolio credit risk. Estimates are presented for stress scenarios of varying severity. Implications of use of different analysis time horizons are analyzed. Results for a numeraire portfolio are quite sensitive to such variations. Although the analysis is framed in terms of recent proposals to revise regulatory capital requirements for banks, the arguments and results are also relevant for bankers making capital structure decisions.
... ν stands for the k-regime variance at period t, j i β α α , , 0 are constants. BIS (2009) recommends a stressed value-at-risk (SVaR), a methodology initially proposed by Kupiec (1998) Where: Max, RC, VaR, SVar and kt stand for Maximum, Required Capital, Value at Risk, Stressed Value at Risk and a factor kt defined by the country financial regulator (usually a Central Bank). The original formula from Kupiec (1998) specifies only the last term, without the multiplier (3+ kt) and a sample number N in place of the fixed 60. ...
... BIS (2009) recommends a stressed value-at-risk (SVaR), a methodology initially proposed by Kupiec (1998) Where: Max, RC, VaR, SVar and kt stand for Maximum, Required Capital, Value at Risk, Stressed Value at Risk and a factor kt defined by the country financial regulator (usually a Central Bank). The original formula from Kupiec (1998) specifies only the last term, without the multiplier (3+ kt) and a sample number N in place of the fixed 60. ...
Are the recommendations from the Bank for International Settlements (BIS) effective to a broad set of financial crises? We submitted two of the main Basel III recommendations for market risk to a back test: the capital requirements and the Value at Risk (VaR) methodology that includes the BIS’s Stressed VaR. We tested the main Brazilian currency exchange (U.S. Dollar to Brazilian Reais) and currency exchange swaps contracts through volatility-based VaR methodologies in the period that comprises the so-called Brazilian confidence crisis, which occurred in the second half of 2002.
While the Stressed VaR revealed inapplicable, due to historical data shortage, the capital requirements level appeared innocuous, due to the high levels of daily volatility – daily oscillation limits may have a significant role on crisis mitigation. To circumvent the lack of either historical information or optimal window for stress patterns, we suggest to calibrate the Stressed VaR or the recently announced Expected Shortfall with a historical VIX (Volatility Index, Chicago Board Options Exchange), working as a volatility scale. We suggest modelling with other densities, apart from the BIS recommended standard normal.
... Stress testing is a term used in financial practice without any generally accepted definition. It appears in the context of quantification of losses or risks that may appear under special, mostly extremal circumstances [19]. Such circumstances are described by certain scenarios which may come from historical experience (a crisis observed in the past)-historical stress test, or may be judged to be possible in the future given changes of macroeconomic, socioeconomic or political factors-prospective stress test, etc. ...
... Asymptotic statistics and a detailed analysis of optimal solutions of parametric quadratic programs may help to derive asymptotic results concerning the "estimated" optimal portfolio composition obtained for an asymptotically normal estimateΣ of Σ. Here we follow a suggestion of [19] and rewrite the variance matrix as Σ = DCD with the diagonal matrix D of "volatilities" (standard deviations of the marginal distributions) and the correlation matrix C. Changes in the covariances may be then modeled by "stressing" the correlation matrix C by a positive semidefinite stress correlation matrixĈ ...
Practical use of the contamination technique in stress testing for risk measures Value at Risk (VaR) and Conditional Value at Risk (CVaR) and for optimization problems with these risk criteria is discussed. Whereas for CVaR its application is straightforward, the presence of the simple chance constraint in the definition of VaR requires that various distributional and structural properties are fulfilled, namely, for the unperturbed problem. These requirements rule out direct applications of the contamination technique in the case of discrete distributions, which includes the empirical VaR. On the other hand, in the case of a normal distribution and parametric VaR one may exploit stability results valid for quadratic programs.
... The purpose of this study is to use real market data to create correlation flows that satisfy the requirements listed above and approach the valid correlation. There are existing solutions available to address the same problem; see [10][11][12][13]. ...
Standard correlation analysis is one of the frequently used methods in financial markets. However, this matrix can give erroneous results in the conditions of chaos, fractional systems, entropy, and complexity for the variables. In this study, we employed the time-dependent correlation matrix based on isospectral flow using the Lie group method to assess the price of Bitcoin and gold from 19 July 2010 to 31 December 2024. Firstly, we showed that the variables have a chaotic and fractional structure. Lo’s rescaled range (R/S) and the Mandelbrot–Wallis method were used to determine fractionality and long-term dependence. We estimated and tested the d parameter using GPH and Phillips’ estimators. Renyi, Shannon, Tsallis, and HCT tests determined entropy. The KSC determined the evidence of the complexity of the variables. Hurst exponents determined mean reversion, chaos, and Brownian motion. Largest Lyapunov and Hurst exponents and entropy methods and KSC found evidence of chaos, mean reversion, Brownian motion, entropy, and complexity. The BDS test determined nonlinearity, and later, the time-dependent correlation matrix was obtained by using the stochastic SO(2) Lie group. Finally, we obtained robustness check results. Our results showed that the time-dependent correlation matrix obtained by using the stochastic SO(2) Lie group method yielded more successful results than the ordinary correlation and covariance matrix and the Spearman correlation and covariance matrix. If policymakers, financial managers, risk managers, etc., use the standard correlation method for economy or financial policies, risk management, and financial decisions, the effects of nonlinearity, fractionality, entropy, and chaotic structures may not be fully evaluated or measured. In such cases, this can lead to erroneous investment decisions, bad portfolio decisions, and wrong policy recommendations.
... As Gillard-Zhigljavsky [3] said, the structured low rank approximation is a difficult optimization problem, so there is much work to be done. In the last few years, there has been a constantly increasing interest in developing the theory and numerical methods for the nearest low rank approximation of a correlation matrix, due to their wide applications in the fiance and risk management [6], machine learning [15], stress testing of bank [13], industrial process monitoring [7] and image processing [5]. Recently, problem (1.1) with m = 1 has been extensively studied, and the research results mainly concentrate on the following two cases. ...
In this paper, we consider the generalized low rank approximation of the correlation matrices problem which arises in the asset portfolio. We first characterize the feasible set by using the Gramian representation together with a special trigonometric function transform, and then transform the generalized low rank approximation of the correlation matrices problem into an unconstrained optimization problem. Finally, we use the conjugate gradient algorithm with the strong Wolfe line search to solve the unconstrained optimization problem. Numerical examples show that our new method is feasible and effective.
... For more details, properties of the VaR and for an overview on the 1.5. Risk Measures topic, one may consult the textbooks of Artzner et al. (1999); Jorion (2000); Kupiec (2002) or Denuit et al. (2005). ...
... The Stress Testing methods were further advanced by the introduction of Artificial Intelligence (AI) and Generative AI technology These advanced technologies allowed for the generation of stress scenarios beyond the traditional predefined adverse scenarios. AI-driven algorithms enabled financial institutions to simulate a wide range of stress scenarios and assess the potential impact on various risk factors [8]. ...
This research article explores the transformative impact of technological advancements on banking stress testing, a critical tool for assessing the resilience of financial institutions in adverse economic scenarios. The article delves into the historical background of stress testing, highlighting its evolution post-2008 financial crisis and the role of regulatory frameworks in shaping its methodologies. It further examines the integration of artificial intelligence (AI), big data, and real-time analytics in enhancing the accuracy and efficiency of stress tests. Practical applications, including real-time monitoring, advanced data analytics, and big data utilization, are discussed alongside strategies for banks to implement these technologies effectively. The article also addresses challenges such as data management, regulatory compliance, and talent acquisition. Finally, it explores future trends in stress testing, emphasizing the role of AI, machine learning, and RegTech in advancing risk management practices. The findings underscore the necessity for financial institutions to adopt technological innovations to strengthen their risk management frameworks and ensure financial stability.
... En la escala de la bondad de las métricas destacan las métricas para el riesgo de mercado, J.P. Morgan (1994), Duffie y Pan (1997), y Vilariño (2016), por tres cuestiones esenciales: i) el riesgo de mercado se mide a un plazo muy corto, 10 días, ii) existe información pública diaria de los instrumentos financieros expuestos al riesgo de cambio de precio, y iii) existen metodologías bien definidas para el contraste de los modelos, Kupiec (1995) y Kupiec (1998). ...
El artículo informa del episodio sufrido por Silicon Valley Bank y los impactos sistémicos que generó el sistema bancario mundial. Se analizan los cambios de la regulación de los bancos comerciales en la normativa denominada Basilea III, a raíz de la gran crisis financiera y las dificultades existentes para la medición de los riesgos que están expuestos los bancos, especialmente en el caso de los riesgos de crédito y de liquidez. Finalmente, se pone el acento en la importancia de la supervisión y en sus grandes dificultades, teniendo en cuenta no sólo las dificultades teóricas sino las dificultades políticas.
... Representing "the worst expected loss over a given horizon under normal market conditions at a given level of confidence", the popularity of VaR is mainly due to its simplicity and ease of computation. The interested reader may consult Artzner et al. (1999), Jorion (2000), Kupiec (2002) and Denuit et al. (2005) for overviews on the topic. ...
... Survey of Literature Cihak (2007) provided stress testing as the process of identification of specific vulnerabilities. The works of Kupiec, 1998 Table 2 describes the literature on stress testing of credit risk and its coverage worldwide. Figure 1 shows the conceptual framework of conducting a macro-stress test. ...
Since the last COP27 meeting on November 18, 2022 and the upcoming meet in 2023, climate change would have become a physical reality. The impact of climate draws attention of regulators to emphasize preparedness of the banking sector. This paper highlights the qualitative factors, mapping between scenarios and outcomes using historical data on food and non food credit losses in the banking sector for a sample of 55 Banks.
... Engle [26] recognized correlations among US financial indices, US bonds, and foreign currency changes over time and proposed a new measure to improve the assessment and forecasting of financial risks during the financial crisis. Kupiec [27] conducted stress tests on US financial institutions to measure the potential losses of these financial institutions. Financial risk spillovers and financial systemic risk are two main issues of concern in the US financial market. ...
The analysis and prediction of systemic financial risks in the US during the COVID-19 pandemic is of great significance to the stability of financial markets in the US and even the world. This paper aims to predict the systemic financial risk in the US before and during the COVID-19 pandemic by using copula–GJR–GARCH models with component expected shortfall (CES), and also identify systemically important financial institutions (SIFIs) for the two comparative periods. The empirical results show that the overall systemic financial risk increased after the outbreak of the COVID-19 pandemic, especially in the first half of the year. We predicted four extreme risks that were basically successful in capturing the high risks in the US financial markets. Second, we identified the SIFIs, and depository banks made the greatest contribution to systemic risk from four financial groups. Third, after the outbreak of the epidemic, the share of Broker–Dealer and Other Institutions in the overall systemic risk has apparently increased. Finally, we recommend that the US financial regulators should consider macro-prudential guidance for major financial institutions, and we should pay more attention to Broker–Dealers, thereby improving the financial stability of the US and the global financial markets.
... Macro stress testing has also been performed by regulators as part of the system-wide stress tests that they themselves have implemented, post the crisis, to assess systemic risk in different countries' banking markets. Berkowitz (1999), Kupiec (1998), Lopez (2005) and Schachter (2001) discuss how stress testing may be used in ways that complement and are more or less integrated with VaR analysis. CGFS (2001), CGFS (2005) and CGFS (2000) present survey information and best practice guidelines on stress testing in financial firms. ...
This paper analyzes the joint distribution of changes in agency credit ratings. We estimate both intra-and inter-industry correlations using Maximum Likelihood techniques. The analysis is performed unconditionally and then conditional on de-trended GDP. The latter estimates may be used for macro stress testing in which the credit quality of a portfolio is simulated conditional on a hypothesized future path of real output.
... This decade is important in the history of systemic risk development, because it spans considerable time frame before and after the Global Financial Crisis (GFC) and, thus, enables discovery of the systemic associative patterns between CEO pay and systemic 1 CoVar measures the change in the value at risk of the whole financial system conditional on the state of distress of financial firm relative to the median state of the firm. Value at risk (VaR) is a common method that measures loss associated with the 1-5% chance occurring events on financial markets (Kupiec 1998;Adrian and Brunnermeier 2016). 2 Dollar CoVar, CoVar $ , is a measure of systemic risk, reflecting the size of a financial firm and is calculated as the size of a firm times the CoVar, that is the change in the value at risk of a financial system conditional on a difference in the state of the firm relative to its median state (Adrian and Brunnermeier 2016). risk through the business cycles that include a couple of recoveries from recessions reflected with the periods of economic growth. ...
We examine the role of chief executive officers’ (CEO) pay in contribution to systemic risk in the USA. In particular, by extending the CoVar model of Adrian and Brunnermeier (Am Econ Rev 106(7):1705–1741, 2016), we document that the systemic risk measure of dollar delta CoVar is positively influenced by CEO pay. Differentiation between the types of CEO pay incentives suggests that bonus and option awards comprise major contribution to systemic risk. It follows that governance measures that are aimed at systemic risk management can benefit from distinguishing between short-term and long-term CEO incentives.
... Stress-testing cannot guarantee the identification of actual impacts on a portfolio of future events, but provides another tool in the risk manager's armoury. Stress-tests are designed to determine how a portfolio might respond to adverse developments, including portfolio allocation, 2 and detecting weak spots early, thus facilitating preventative action, typically focusing on key risks such as market risk, credit risk and liquidity risk. 3 Stress-testing covers a range of methodologies. 4 For current purposes it is sufficient to regard stress-tests as being either based on historical data ('historical stress-tests') or invented scenarios ('artificial stress-tests'). ...
Attempting to put meaningful numbers to portfolio risks is challenging. Conventional risk measures are considered often not to fully capture all risks inherent in a portfolio, particularly under difficult market conditions. Under such conditions stress-testing against artificial scenarios may help identify and quantify risks within a portfolio. Stress-tests also help reassure a portfolio or risk manager as to how a portfolio might respond to specific concerns. This paper investigates an example of stress-testing a portfolio of conventional assets against market risks using artificial scenarios based around changes to the portfolio variance-covariance matrix. Hypothetical variance-covariance matrix stress-tests include making changes to correlations between assets to explore impacts on portfolio risks. Portfolio correlations, however, cannot be changed arbitrarily to reflect a risk manager’s concerns without running the risk of implausible stressed returns and variance-covariance matrices that are not positive semi-definite. Different methods have been proposed in the literature to overcome this. This paper applies two such methods to a portfolio of four assets with the aim of illustrating the processes involved as well as drawing out differences in the approaches, enabling a discussion of their strengths and weaknesses.
... [36] suggest a conditional density forecast in discrete time, but their approach is also limited to a factor model assumption. Other papers involving general conditional modelling in a scenario generation setting are given by [37], who analyze ways to model expert judgement as combined probability distributions but provide no testing or inclusion in a forecasting framework; [38], who extend the Black Litterman model by forecasting asset distributions conditionally on both views for its mean and variance but assume a linear Gaussian model for the asset returns; or [8], who propose a Market-Driven Scenario Approach based on [39] for forecasting financial assets conditional on fixed values for some of the considered assets and evaluate it on the P&L distribution of a portfolio given forecasted Brexit scenarios. However, their approach does not consider a multi period time horizon. ...
We introduce the notion of Point in Time Economic Scenario Generation (PiT ESG) with a clear mathematical problem formulation to unify and compare economic scenario generation approaches conditional on forward looking market data. Such PiT ESGs should provide quicker and more flexible reactions to sudden economic changes than traditional ESGs calibrated solely to long periods of historical data. We specifically take as economic variable the S&P500 Index with the VIX Index as forward looking market data to compare the nonparametric filtered historical simulation, GARCH model with joint likelihood estimation (parametric), Restricted Boltzmann Machine and the conditional Variational Autoencoder (Generative Networks) for their suitability as PiT ESG. Our evaluation consists of statistical tests for model fit and benchmarking the out of sample forecasting quality with a strategy backtest using model output as stop loss criterion. We find that both Generative Networks outperform the nonparametric and classic parametric model in our tests, but that the CVAE seems to be particularly well suited for our purposes: yielding more robust performance and being computationally lighter.
... SeeKupiec (2002) andJorion (2007) for overviews.9 Since we focus on the left-tail risk, we set q to be 1%. ...
We investigate the dependency, risk spillovers, and systemic risk between the sectoral indices returns of the Bombay stock exchange (BSE) and oil prices using recently developed empirical techniques. The dependence is modelled using the time varying Stochastic Autoregressive Copulas (SCAR). Conditional value-at-risk (CoVaR), ΔCoVaR and marginal expected shortfall (MES) measures are used to examine the systemic risk. We find rotated Gumbel and normal copulas to be the best fitting in our analysis. Sectors such as energy, power, and industrial exhibit higher persistence in dependence structure compared to other sectors. Our results reveal that the underlying forces of the dependence between oil prices with other industries vary across time, albeit not so much during stable periods, but increase remarkably during turbulent times. All sectors are affected significantly by extreme oil price movements. The average short-run MES is highest for the metals, materials, and industrials sectors. The lowest average short-run MES values are observed for the fast-moving consumer goods, auto, and carbon sectors. Our risk analysis results reveal that Indian stock sectors are not resistant to oil shocks and there exists significant systemic risk between these markets and the crude oil market.
... There are already methods available that were designed to tackle the same problem, see e.g., [1][2][3][4]. Newton-based methods for approximating covariance matrices can be found in [5][6][7]. Furthermore, there exist methods using hyperspherical decomposition [8] and unconstrained convex optimization [9]. ...
Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.
... We estimate median (Expected) and 95th percentile (Value at Risk, VaR) annual damage for the sake of comparison with other published work, but we assess the risk of low-probability sea-level rises (Hull, 2018;Wilmott, 2014) using the Expected Shortfall (ES(95%)), which is the mean loss when the 95% percentile is exceeded. This risk measure is commonly used in financial economics to stress test systems and identify risk thresholds (Kupiec, 1998). ...
The high degree of uncertainty associated with the extent of future sea-level rise stems primarily from the potential mass loss of the Greenland and Antarctica ice-sheets. We explore the impact of this uncertainty on economic damage due to sea-level rise for 136 major coastal cities. We compare the probability distribution for damage under the assumption of no adaptation for two relative sea-level projections: the RCP 8.5 scenario from the IPCC Fifth Assessment Report and a High-end scenario that incorporates expert opinion on additional ice-sheet melting. We use the 50th and 95th percentiles to estimate expected damage and one risk measure, the Expected Shortfall ES (95%), which represents the impact of low-probability, high-damage coastal flood risk (above the 95th percentile). Aggregate expected damage by 2050 under RCP 8.5 is US3,082 billion. Under the High-end scenario, ES(95%) figures in Guangzhou and New Orleans by 2050 are twice as high as the expected damage. The city of Guangzhou leads the ranking under both scenarios, followed by Mumbai and New Orleans. Our results suggest that it is critical to incorporate the possibility of High-end scenarios into coastal adaptation planning for future sea-level rise, especially for risk-averse decision-making.
... Golub et al. (2018) propose a framework for calibrating asset returns to financial scenarios. Their Market-Driven Scenario (MDS) approach follows the conditioning philosophy outlined by Kupiec (2002). The core concept is to consider the joint distribution of factors that drive financial outcomes, and look at the conditional distribution of outcomes given an explicit value for a subset of these factors that capture the scenario. ...
Economic Scenario Generators (ESGs) simulate economic and financial variables forward in time for risk management and asset allocation purposes. It is often not feasible to calibrate the dynamics of all variables within the ESG to historical data alone. Calibration to forward-information such as future scenarios and return expectations is needed for stress testing and portfolio optimization, but no generally accepted methodology is available. This paper introduces the Conditional Scenario Simulator, which is a framework for consistently calibrating simulations and projections of economic and financial variables both to historical data and forward-looking information. The framework can be viewed as a multi-period, multi-factor generalization of the Black-Litterman model, and can embed a wide array of financial and macroeconomic models. Two practical examples demonstrate this in a frequentist and Bayesian setting.
... However, as commented in [Rebonato and Jäckel, 2000], the drawback is that other portions of the matrix can be changed in an uncontrolled fashion. The shrinkage method proposed by Kupiec in [Kupiec, 1998] has the main drawback that "there is no way of determining to what extent the resulting matrix is optimal in any easily quantifiable sense", see [Rebonato and Jäckel, 2000]. Furthermore, the hyperspherical decomposition method and the unconstrained convex optimization approach are proposed in [Rebonato and Jäckel, 2000] and [Qi and Sun, 2010], respectively. ...
In many areas of finance and of risk management it is interesting to know how to specify time-dependent correlation matrices. In this work we propose a new methodology to create valid time-dependent instantaneous correlation matrices, which we called correlation flows. In our methodology one needs only an initial correlation matrix to create these correlation flows based on isospectral flows. The tendency of the time-dependent matrices can be controlled by requirements. An application example is presented to illustrate our methodology.
... A key feature of these exercises was to shift away from previous "value at risk" (VaR) approaches. VaR analysis can offer some substantial advantages, including its practical viability and conceptual attractiveness (Kupiec, 1998) and the ability to contrast multiple models and calibrations (see for example Alexander and Sheedy, 2008). But with its decline, stress tests instead became increasingly reliant on a form of scenario analysis: taking unexpected (downside) macro scenarios and estimating how those impact, via loan and securities losses, on bank capital. ...
This paper describes an approach for stress testing banks that is consistent across economies and geographies, in contrast to common “macro scenario” driven approaches. The latter would require economic scenarios to be both equally likely (in a probabilistic sense) and equally stressful (in a conditional loss sense) across countries in order to be comparable. The paper proposes a three-pronged approach for stressing bank solvency, which incorporates recalibrating pre-crisis Basel capital assumptions, adapting the BIS “expected shortfall” approach for securities, and using granular data for income haircuts. Loan losses are quantified using a simple “multiples” approach, starting from expected outcomes, which is derived from the pre-crisis Basel technical proposal. The approach is practical, can be more granular or conducted at a high level, depending on data availability, and offers a simple way for regulators, investors or risk assessors to compare and contrast stresses in different banking systems. Of the eight bank defaults recorded globally during 2017, this approach would have given a better “rank ordering” for seven of them, indicating the approach adds value to traditional solvency metrics.
... This study focuses on the most common measure of tail risk, Value at risk (VaR), which is defined as the worst-case scenario in terms of losses during a specific period. Overviews of VaR can be found in Kupiec (2002) and Jorion (2007). Due to its convenience, VaR is a popular method for measuring tail risk. ...
... Following e.g. Kupiec (1998) , we define a stress scenario on one set of ("core") risk factors and, assuming that the given covariance matrix is unaltered by the stress scenario, set the remaining ("peripheral") risk factors to their optimal estimates conditional on the scenario. Let β s denote the j < m core factor parameters that are stressed directly. ...
In 2012, JPMorgan accumulated a USD 6.2 billion loss on a credit derivatives portfolio, the so-called “London Whale”, partly as a consequence of de-correlations of non-perfectly correlated positions that were supposed to hedge each other. Motivated by this case, we devise a factor model for correlations that allows for scenario-based stress testing of correlations. We derive a number of analytical results related to a portfolio of homogeneous assets. Using the concept of Mahalanobis distance, we show how to identify adverse scenarios of correlation risk. In addition, we demonstrate how correlation and volatility stress tests can be combined. As an example, we apply the factor-model approach to the “London Whale” portfolio and determine the value-at-risk impact from correlation changes. Since our findings are particularly relevant for large portfolios, where even small correlation changes can have a large impact, a further application would be to stress test portfolios of central counterparties, which are of systemically relevant size.
... However, as commented in [Rebonato and Jäckel, 2000], the drawback is that other portions of the matrix can be changed in an uncontrolled fashion. The shrinkage method proposed by Kupiec in [Kupiec, 1998] has the main drawback that "there is no way of determining to what extent the resulting matrix is optimal in any easily quantifiable sense", see [Rebonato and Jäckel, 2000]. Furthermore, the hyperspherical decomposition method and the unconstrained convex optimization approach are proposed in [Rebonato and Jäckel, 2000] and [Qi and Sun, 2010], respectively. ...
In many areas of finance and of risk management it is interesting to know how to specify time-dependent correlation matrices. In this work we propose a new methodology to create valid time-dependent instantaneous correlation matrices, which we called correlation flows. In our methodology one needs only an initial correlation matrix to create these correlation flows based on isospectral flows. The tendency of the time-dependent matrices can be controlled by requirements. An application example is presented to illustrate our methodology.
... Financial risk is a challenge for most business firms. This is often due to the lack of necessary resources, with regards to manpower, databases and specialty of knowledge to perform a standardized and structured risk management (Kupiec 1998). Smaller firms do not Corresponding author email: solomonenimu@gmail.com ...
This study assessed the financial risk of food vendors in Calabar Metropolis. It specifically sought to identify the types of financial risk in the study area, the level of financial risk, the effect of financial risk factors on vendor’s sales and strategies used to manage financial risk. The study used random sampling technique to select 120 restaurants in Calabar metropolis. Data were obtained from primary source using structured questionnaire and analyzed using descriptive and inferential statistics. The results showed that interest rate risk was the most common type of financial risk in the study area at 46.7%. The result also revealed that the mean financial risk level was 34.93%. Three variables were statistically significant in influencing sales and these were taxation, variable cost and financial risk. The result further revealed that savings was the most common financial risk management technique at 37.2%. The study therefore, recommends that food vendors should insure their businesses to reduce the effects of financial risk. Food vendors should maintain a balance with lending institutions to curtail financial risk for their viability and sustainability.
... Additionally to the estimated bad correlation matrix C one chooses a target correlation matrix C 0 (as reference, see e.g. Kupiec, [7]). Restricting the solution of (GCP) to the line between C and C 0 , problem (GCP) becomes in this case ...
... Following e.g. Kupiec (1998), we define a stress scenario on one set of ("core") risk factors and, assuming that the given covariance matrix is unaltered by the stress scenario, set the Proposition 3). All graphs show a 99% Value-at-Risk, and the initial and unstressed β is calibrated to an average asset correlation of ρ ≈ 0.3, unless indicated otherwise (cf. ...
In 2012, JPMorgan accumulated a USD 6.2 billion loss on a credit derivatives portfolio, the so-called "London Whale", partly as a consequence of de-correlations of non-perfectly correlated positions that were supposed to hedge each other. Motivated by this case, we devise a factor model for correlations that allows for scenario-based stress-testing of correlations. We derive a number of analytical results related to a portfolio of homogeneous assets. Using the concept of Mahalanobis distance, we show how to identify adverse scenarios of correlation risk. As an example, we apply the factor-model approach to the "London Whale" portfolio and determine the value-at-risk impact from correlation changes. Since our findings are particularly relevant for large portfolios, where even small correlation changes can have a large impact, a further application would be to stress-test portfolios of central counterparties, which are of systemically relevant size.
... The two risk measures proposed in this article can be used in conjunction with the concept of ALR to decide on appropriate adaptation. Indeed, these measures are very appropriate for stress testing in an analogous way to the tests done in the financing and banking system to assess resilience (Kupiec 1998). These tests consist of assessing whether a system can or cannot recover (or how much effort will require to recover) from certain negative events occurring. ...
This addendum adds to the analysis presented in 'Understanding risks in the light of uncertainty: low-probability, high-impact coastal events in cities' Abadie et al (2017 Environ. Res. Lett. 12 014017). We propose to use the framework developed earlier to enhance communication and understanding of risks, with the aim of bridging the gap between highly technical risk management discussion to the public risk aversion debate. We also propose that the framework could be used for stress-testing resilience.
... Financial risk is a challenge for most business firms. This is often due to the lack of necessary resources, with regards to manpower, databases and specialty of knowledge to perform a standardized and structured risk management (Kupiec 1998). Smaller firms do not Corresponding author email: solomonenimu@gmail.com ...
Abstract This study assessed the financial risk of food vendors in Calabar Metropolis. It specifically sought to identify the types of financial risk in the study area, the level of financial risk, the effect of financial risk factors on vendor’s sales and strategies used to manage financial risk. The study used random sampling technique to select 120 restaurants in Calabar metropolis. Data were obtained from primary source using structured questionnaire and analyzed using descriptive and inferential statistics. The results showed that interest rate risk was the most common type of financial risk in the study area at 46.7%. The result also revealed that the mean financial risk level was 34.93%. Three variables were statistically significant in influencing sales and these were taxation, variable cost and financial risk. The result further revealed that savings was the most common financial risk management technique at 37.2%. The study therefore, recommends that food vendors should insure their businesses to reduce the effects of financial risk. Food vendors should maintain a balance with lending institutions to curtail financial risk for their viability and sustainability. Keywords: Financial Risk; Food Vendors; Risk Management; Calabar Metropolis
... VaR analysis offers some substantial advantages, such as its practical viability and conceptual attractiveness, as presented by Kupiec (1998) among others from a historical context, and the ability to consider and contrast multiple models and calibrations, for Source: Adapted from "Stress testing the UK banking system: key elements of the 2014 stress test" by Bank of England (2014, April). Retrieved from http://www.bankofengland.co.uk/financialstability/Documents/fpc/keyelements.pdf ...
Forecasts, models and stress tests are important tools for policymakers and business planners. Recent developments in these related spheres have seen greater emphasis placed on stress tests from a regulatory perspective, while at the same time forecasting performance has been criticized. Given the interlinkages between the two, similar limitations apply to stress tests as to forecasts and should be borne in mind by practitioners. In addition, the recent evolution of stress tests, and in particular the increasing popularity of scenario-based approaches, raises concerns about how well the shortcomings of the associated models are understood. This includes estimated stress cases relative to base cases – the degree of pain – that simple scenario modelling approaches engender. This paper illustrates this phenomenon using simulation techniques and demonstrates that more extreme stress scenarios need to be employed in order to match the inference from simple value-at-risk approaches. Alternatively, complex modelling approaches can address this concern, but are not widely used to date. Some policymakers seem to be aware of these issues, judging by the severity of some recent stress scenarios. © 2017, University of Finance and Management in Warsaw. All rights reserved.
... The appearance of stress tests in the mathematical finance literature, on the other hand, is relatively new and there is yet no unified theory of stress testing. The foundations of the link between stress tests and risk models started with Kupiec (1998) who examined crossmarket effects resulting from a market shock. In a seminal paper, Berkowitz (2000) for the first time came up with the idea of folding stress tests into a risk model, thereby assigning all scenarios' certain probabilities. ...
Purpose
Adverse developments in the global finance industry clearly underlined the importance of stress-testing. One of the key takeaways was the need to strengthen the coverage of the capital framework. Cognisant of this fact, Basel III encapsulates provisions to enhance the financial sector's ability to withstand shocks arising from possible stress events, thereby reducing adverse spillovers into the real economy. Similarly, the IFSB requires Islamic financial institutions to run stress tests as part of capital planning.
Design/methodology/approach
We perform thorough backtests on Islamic and conventional portfolios under widely used risk models, which are characterised by an underlying conditional volatility framework and distribution, to identify the most suitable risk model specification. Associated with an appropriate initial shock and estimation window size, the paper also conducts a model-based stress-test to examine whether the stress losses estimated by the selected models compare favourably to the historical shocks.
Findings
The results suggest that the model-based framework, when combined with an appropriate risk model and distribution, can successfully reproduce past stress periods. The conditional empirical risk model is the most effective one in both long and short portfolio cases particularly when combined with a long-enough estimation window. The relative performance of normal vs. heavy-tailed distributions and symmetric vs. asymmetric risk models, on the other hand, is highly dependent on whether the portfolio is long or short. Finally, we find that the Islamic portfolio is generally associated with lower historical stress losses as compared to the conventional portfolio.
Originality/value
The model-based framework eliminates some of the key problems associated with traditional scenario-based approaches and is easily adaptable to Islamic finance.
The correlation problem is a central focus in statistical analysis and data science, as it aims to quantify the relationships between variables. This paper explores efficient methods for approximating correlation matrices to assess loan eligibility for bank customers. We propose a novel algorithm that utilizes advanced optimization techniques to minimize the difference between actual noisy matrices and approximated correlation matrices. The algorithm is designed for large-scale correlation matrices, such as those with thousands of variables, and employs the interior point primal-dual path-following method. We provide a comprehensive comparative analysis of our methods and the commonly used modified alternating projection method, evaluating their efficacy and computational efficiency based on numerical results.
The Value-at-Risk (VaR) concept was introduced by the American bank JP Morgan at the start of the 1990s to summarize the market risk impacting a portfolio or an assets-and-liabilities position in a single measure with a direct interpretation. The VaR quantifies, within a specified confidence level (typically 95 % or 99 %) the potential loss which could be sustained by a given isolated position, an entire portfolio, or a bank as a whole, in a short period of time (typically from 1 to 10 trading days) in normal market conditions. Whereas the VaR is merely a quantile of the distribution of losses (Sect. 27.1), calculating it may turn out to be complicated for positions that include many different instruments, among them derivatives (Sect. 27.2). Furthermore, the VaR has various shortcomings, and other indicators such as Expected Shortfall (Sect. 27.3) and risk measuring tools (Sect. 27.4) have been developed to overcome these deficiencies.
With political and economic scenarios changing at an ever faster pace, it is necessary to understand the potential effects on asset prices. Today, the topic of rising inflation in the US as well as in the Eurozone, although still considered temporary by central banks, confronts us with the "unexpected risk" of a deviation from the baseline scenario. This implies the risk of having an aggressive monetary policy in the US, in a restrictive direction, therefore harmful to the financial markets. In this context, the question arises: is it possible to contemplate these events beforehand and act in good time? The answer is Yes and good risk management practices are important, using stress testing / scenario analysis techniques to accompany risk measures such as VaR and Expected Shortfall. Implementing this concept, through the implementation of stress test / scenario analysis - Bloomberg Economics Forecast Models® and Bloomberg Factor Models® - the present work seeks to consider plausible adverse scenarios that may arise and to assess the related impacts in terms of portfolio. The final aim is to improve the information set for the investor, allowing him to avoid potential market falls, as far as possible, that could prevent him from achieving his investment objectives.
There is a large variability in profitability and productivity between farms operating with automatic milking systems (AMS). The objectives of this study were to identify the physical factors associated with profitability and productivity of pasture-based AMS and quantify how changes in these factors would affect farm productivity. We utilised two different datasets collected between 2015 and 2019 with information from commercial pasture-based AMS farms. One contained annual physical and economic data from 14 AMS farms located in the main Australian dairy regions; the other contained monthly, detailed robot-system performance data from 23 AMS farms located across Australia, Ireland, New Zealand, and Chile. We used linear mixed models to identify the physical factors associated with different profitability (Model 1) and partial productivity measures (Model 2). Additionally, we conducted a Monte Carlo simulation to evaluate how changes in the physical factors would affect productivity. Our results from Model 1 showed that the two main factors associated with profitability in pasture-based AMS were milk harvested/robot (MH; kg milk/robot per day) and total labour on-farm (full-time equivalent). On average, Model 1 explained 69% of the variance in profitability. In turn, Model 2 showed that the main factors associated with MH were cows/robot, milk flow, milking frequency, milking time, and days in milk. Model 2 explained 90% of the variance in MH. The Monte Carlo simulation showed that if pasture-based AMS farms manage to increase the number of cows/robot from 54 (current average) to ∼ 70 (the average of the 25% highest performing farms), the probability of achieving high MH, and therefore profitability, would increase from 23% to 63%. This could make AMS more attractive for pasture-based systems and increase the rate of adoption of the technology.
In the shipping market, ports are increasingly interconnected in the global container network. A port disruption in one place can lead to a chain effect on subsequent ports. Nowadays, China's container port throughput accounts for around 30% of the global port throughput, manifesting its essential role in the global container trade system. Therefore, any disruption in Chinese ports could have a cascading effect on the world container trade. This study explores the potential impact on the container main trading lanes given a stress scenario for a number of Chinese ports. We are particularly interested in the degree of risk spillover to the different trading lanes. The stress scenario used in this study is a severe drop in the container throughput in major Chinese ports. This drop will be reflected in quantiles of the respective throughput distributions that are close to zero. D-vine copula-based quantile regression is used to measure the throughput impact of stressed Chinese ports as covariates on other response container trading routes. Copula is a powerful tool to measure the tail dependence structure between throughputs of major Chinese ports and major container trading routes. Results show that port disruptions in China severely impact the major container trading routes. The Asia to Sub Saharan Africa route is extremely vulnerable to downside risk given any port disruption event in China. We have also compared the results with the historical trade and port call data. The results provide significant implications for container shipping risk management.
This companion book contains the solutions of the tutorial exercises which are included in the Handbook of Financial Risk Management.
The table of contents is the following:
1. Introduction.
Part I Risk Management in the Financial Sector
2. Market Risk.
3. Credit Risk.
4. Counterparty Credit Risk and Collateral Risk.
5. Operational Risk.
6. Liquidity Risk.
7. Asset Liability Management Risk.
8. Systemic Risk and Shadow Banking System.
Part II Mathematical and Statistical Tools
9. Model Risk of Exotic Derivatives.
10. Statistical Inference and Model Estimation.
11. Copulas and Dependence Modeling.
12. Extreme Value Theory.
13. Monte Carlo Simulation Methods.
14. Stress Testing and Scenario Analysis.
15. Credit Scoring Models.
Conclusion
Appendix
A.1 Numerical Analysis.
A.2 Statistical and Probability Analysis.
A.3 Stochastic Analysis.
The covariance matrix of asset returns can change drastically and generate huge losses in portfolio value under extreme conditions such as market interventions and financial crises. Estimation of the covariance matrix under a chaotic market is often a call to action in risk management. Nowadays, stress testing has become a standard procedure for many financial institutions to estimate the capital requirement for their portfolio holdings under various stress scenarios. A possible stress scenario is to adjust the covariance matrix to mimic the situation under an underlying stress event. It is reasonable that when some covariances are altered, other covariances should vary as well. Recently, Ng et al. proposed a unified approach to determine a proper correlation matrix which reflects the subjective views of correlations. However, this approach requires matrix vectorization and hence it is not computationally efficient for high dimensional matrices. Besides, it only adjusts correlations, but it is well known that high correlations often go together with high standard deviations during a crisis period. To address these limitations, we propose a Bayesian approach to covariance matrix adjustment by incorporating subjective views of covariances. Our approach is computationally efficient and can be applied to high dimensional matrices.
The objective of this paper was to analyze the risk management of a portfolio composed by Petrobras PN, Telemar PN and Vale do Rio Doce PNA stocks. It was verified if the modeling of Value-at-Risk (VaR) through the place Monte Carlo simulation with volatility of GARCH family is supported by hypothesis of efficient market. The results have shown that the statistic evaluation in inferior to dynamics, evidencing that the dynamic analysis supplies support to the hypothesis of efficient market of the Brazilian share holding market, in opposition of some empirical evidences. Also, it was verified that the GARCH models of volatility is enough to accommodate the variations of the shareholding Brazilian market, since the model is capable to accommodate the great dynamic of the Brazilian market.
Abstract
Stress testing is a simulation technique to evaluate portfolio reactions to several critical situations. In this paper, we review different stress testing methodologies to examine impacts of different stress scenarios on an Iranian equity portfolio. We identify the extreme tails of all risk factors in our portfolio by extreme value theory and model their dynamic and nonlinear dependence structures with copula functions. We performed three stress tests such as historical, hybrid and hypothetical stress scenarios to simulate the joint evolution of risk factors over time in a realistic way. According to the empirical findings, we find that historical scenario method is not a suitable tool for stress testing due to several drawbacks and show the importance of forward-looking analysis such as hybrid and hypothetical scenarios. We also indicate that the hypothetical stress approach is superior to the other two scenarios from the perspective of stress testing.
Keywords
Stress testing; Value at Risk; Expected Shortfall; Extreme Value Theory; t Copula; kernel smoothed empirical distribution
Contrary to the common approach of stress-testing under which banks are evaluated whether they are distressed, this empirical study chooses to move from the micro stress test approach to a wider new macro stress test category. By being able to stress testing the entire economy of the Eurozone, it will permit big banks to fail and, at the same time, will open room for new banking players to enter the sector, promoting the essence of a healthy destruction. The analysis performs a battery of stress tests, by implementing VaR, Cornish-Fisher VaR, Monte Carlo VaR, Expected Shortfall, Cornish-Fisher Expected Shortfall, and Monte Carlo Expected Shortfall. At the same time, it explicitly considers the new regulatory approach of IFRS9 to incorporate extreme values from forecasted series in the distributions. The analysis also performs two versions of stress tests, one including TARGET2 and one without it. The results document that future stress tests should include TARGET2 values in order to capture a better picture of the stressed economy. The findings from these stress tests clearly illustrate that although there has been a trough after the distress call of 2008, this trough ended. These are results derived without including the TARGET2 transfers. By including the TARGET2 transfers we receive a different picture that possibly acts as a protective mechanism against any future crisis. Caution is still advised, possibly due to some lingering imbalances within the Eurozone.
The chapter briefly reviews the case for stress-testing risk models and recognizes the pressing ‘engineering’ problems that stand between the concept of stress testing and actually doing so for a risk model, such as a portfolio that includes a supposedly optimal allocation of assets. The chapter argues that the application of the Bayesian net ‘technology’ to stress testing introduced in the last decade lends itself particularly well to the need for a practical way to stress-test risk models. The chapter presents proposed solutions to the challenges of stress testing a model with particular reference to the use of Bayesian nets.
As, in light of the recent financial crises, stress tests have become an integral part of risk management and banking supervision, the analysis and understanding of risk model behaviour under stress has become ever more important. In this paper, we present a general approach to implementing stress scenarios in a multi-factor credit portfolio model and analyse asset correlations, default probabilities and default correlations under stress. We use our results to study the implications for credit reserves and capital requirements and illustrate the proposed methodology by stressing a large investment banking portfolio. Although our stress testing approach is developed in a particular credit portfolio model, the main concept - stressing risk factors through a truncation of their distributions - is independent of the model specification and can be applied to other risk types as well.
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