Dynamic CoVaR Modeling and Estimation

This study focuses on the credit risk of Australian financial institutions relative to that of the US. These two countries are chosen because the study is undertaken in Australia, and because Australia is widely considered to have fared far better than the US during the Global Financial (GFC) in terms of both share market volatility and credit defaults, our comparison involves two countries experiencing very different circumstances. The key questions addressed by the paper are firstly, the extent to which the credit risk of Australian financial institutions compares favourably (or otherwise) to the US, and secondly whether credit risk of financial institutions increases (decreases) in a similar fashion over varying time periods. As part of the analysis we will look at a number of aspects. Firstly we will examine the relative financial institution capital levels of the two countries. Although bank capital covers a number of different risks, credit risk is an extremely important component of capital adequacy, and higher credit risk should be reflected in higher capital levels. Secondly we will examine relative levels of credit risk for these countries using non- performing loans and fluctuating asset values, applying various metrics, including Value at Risk (VaR), Conditional Value at Risk (CVaR) and quantile regression. Following the Global Financial Crisis, there has been much criticism leveled at risk management techniques which measure volatility below a specified threshold. One such technique is Value at Risk (VaR). A major criticism is VaR says nothing of the risk beyond that threshold. Conditional Value at Risk (CVaR), on the other hand, measures extreme risk, those risks beyond VaR. Quantile regression divides a dataset into parts, allowing the extreme quantiles to be isolated and measured. Using these techniques, we compare the credit risk of two data sets (US and Australia) over an eleven year time period from 2004 to 2014. This period includes a range of economic circumstances, spanning pre-GFC, GFC and post-GFC. For credit risk we use non-performing loans, as well as a Merton type model which measures volatility in the market asset values of borrowers. We derive a beta which measures credit risk relative to a benchmark. We then compare relative beta changes over time for the two countries. There are a number of important elements and findings highlighted by the paper. Firstly, on an absolute capital basis (equity to assets), as measured by the World Bank, Australian financial institutions have low capital in comparison to their global peers, with a capital ratio that is about half that of US banks. However, the ratio improves substantially on a risk-weighted basis (per the Basel approach), to one that is much closer to US Banks. Australian Banks have a very high home loan component, with home loans attracting a low risk weighting for capital adequacy requirements. Thus there is a much bigger differential between absolute capital ratios and risk weighted capital ratios for Australia than for the US. Secondly the credit risk of Australian financial institutions as measured by the World Bank for non-performing loans is very low in relation to global banks, and is about half that of the US. Thirdly, when we apply measurements such as VaR, CVaR and quantlile regression to non-performing assets and conduct a Beta analysis to measure fluctuations in credit risk, we find that risk for Australian financial institutions moves in line with that of the US. During the GFC, the risk for Australia increased by very similar levels to that of the US, although off a much smaller base. The findings can be important to banks and regulators in understanding credit risk in these countries as well as choosing modelling techniques which are able to measure extreme risk and respond to changing economic circumstances, and thus provide early warning signs of changes in credit risk.

Differential equations connecting VaR and CVaR

The average value at risk (AVaR) is a risk measure which is a superior alternative to value at risk (VaR). In this paper, we present the average value at risk method for fuzzy risk analysis. Firstly, we put forward the new concept of the average value at risk based on credibility theory. Next, we examine some properties of the proposed average value at risk. Then, a kind of fuzzy simulation algorithm is given to calculate the average value at risk. Finally, numerical example is provided. The proposed average value at risk can be applied in many real problems of fuzzy risk analysis.

The Conditional Value-at-Risk (CoVaR) has been proposed by Adrian and Brunnermeier (Am Econ Rev 106:1705–1741, 2016) to measure the impact of a company in distress on the Value-at-Risk (VaR) of the financial system. We propose an extension of the CoVaR, that is, the Conditional Quantile-Located VaR (QL-CoVaR), that better deals with tail events, when spillover effects impact the stability of the entire system. In fact, the QL-CoVaR is estimated by assuming that the financial system and the individual companies simultaneously lie in the left tails of their distributions.

The Mekong—applications of value at risk (VaR) and conditional value at risk (CVaR) simulation to the benefits, costs and consequences of water resources development in a large river basin

Minimizing CVaR and VaR for a portfolio of derivatives

Connectedness and systemic risk between FinTech and traditional financial stocks: Implications for portfolio diversification

The Conditional Value-at-Risk (CoVaR) proposed by Adrian and Brunnermeier (2016) - which quantifies the impact of a company in distress on the Value-at-Risk (VaR) of the financial system - has established itself as a reference measure of systemic risk. In this study, we extend the CoVaR along two dimensions, which lead respectively to: i) the Conditional Autoregressive VaR (CoCaViaR), in which we include autoregressive components of conditional quantiles to explicitly capture volatility clustering and heteroskedasticity; ii) the Conditional Quantile Located VaR (QL-CoVaR), which accentuates the degree of distress in the connections between the conditioning companies and the financial system, as the parameters are estimated by directly linking the left tails of their returns' distributions. By combining the two new risk measures, we also build the Conditional Autoregressive Quantile-Located VaR (QL-CoCaViaR) and introduce a new backtesting method. A large empirical analysis highlights the validity of such approaches and critically discuss their pros and cons. In particular, including quantile-located relationships leads to relevant improvements in terms of predictive accuracy during stressed periods and, therefore, provides a valuable tool for regulators to assess systemic events.

Portfolio risk shows the large deviations in portfolio returns from expected portfolio returns. Value at Risk (VaR) is one method for determining the maximum risk of loss of a portfolio or an asset based on a certain probability and time. There are three methods to estimate VaR, namely variance-covariance, historical, and Monte Carlo simulations. One disadvantage of VaR is that it is incoherent because it does not have sub-additive properties. Conditional Value at Risk (CVaR) is a coherent or related risk measure and has a sub-additive nature which indicates that the loss on the portfolio is smaller or equal to the amount of loss of each asset. CVaR can provide loss information above the maximum loss. Estimating portfolio risk from the CVaR value using Monte Carlo simulation and its application to PT. Bank Negara Indonesia (Persero) Tbk (BBNI.JK) and PT. Bank Tabungan Negara (Persero) Tbk (BBTN.JK) will be discussed in this study. The daily closing price of each BBNI and BBTN share from 6 January 2019 to 30 December 2019 is used to measure the CVaR of the two banks' stock portfolios with this Monte Carlo simulation. The steps taken are determining the return value of assets, testing the normality of return of assets, looking for risk measures of returning assets that form a normally distributed portfolio, simulate the return of assets with monte carlo, calculate portfolio weights, looking for returns portfolio, calculate the quartile of portfolio return as a VaR value, and calculate the average loss above the VaR value as a CVaR value. The results of portfolio risk estimation of the value of CVaR using Monte Carlo simulation on PT. Bank Negara Indonesia (Persero) Tbk and PT. Bank Tabungan Negara (Persero) Tbk at a confidence level of 90%, 95%, and 99% is 5.82%, 6.39%, and 7.1% with a standard error of 0.58%, 0.59%, and 0.59%. If the initial funds that will be invested in this portfolio are illustrated at Rp 100,000,000, it can be interpreted that the maximum possible risk that investors will receive in the future will not exceed Rp 5,820,000, Rp 6,390,000 and Rp 7,100,000 at the significant level 90%, 95%, and 99%

Normal mixture distributions model has been successfully applied in financial time series analysis. In this paper, we estimate the return distribution, value at risk (VaR) and conditional value at risk (CVaR) for monthly and weekly rates of returns for FTSE Bursa Malaysia Kuala Lumpur Composite Index (FBMKLCI) from July 1990 until July 2010 using the two component univariate normal mixture distributions model. First, we present the application of normal mixture distributions model in empirical finance where we fit our real data. Second, we present the application of normal mixture distributions model in risk analysis where we apply the normal mixture distributions model to evaluate the value at risk (VaR) and conditional value at risk (CVaR) with model validation for both risk measures. The empirical results provide evidence that using the two components normal mixture distributions model can fit the data well and can perform better in estimating value at risk (VaR) and conditional value at risk (CVaR) where it can capture the stylized facts of non-normality and leptokurtosis in returns distribution.

For a given time series of daily losses that display volatility clustering, the exact next-day and ten-day value-at-risk (VaR) and expected shortfall (ES) are unknown. The usual procedure is to approximate these values by replacing true parameter values with estimates in the formulas for VaR and ES. Parameter estimation errors for a GARCH (1,1) model for this time series lead to approximate VaR and ES that differ from the exact VaR and ES, respectively. Accurate estimation of the VaR and ES is very important for the proper management of financial risks. In this paper, we find linear regression models in which the response variable is the approximate VaR (ES) and the explanatory variable is the exact VaR (ES). We use these linear regression models to determine the properties of the approximate VaR (ES), conditional on the corresponding exact value. For a given value of the exact VaR (ES), the approximate VaR (ES) is close to being an unbiased estimator of the corresponding exact value, but it may differ from this exact value by more than 10% of the exact value with substantial probability.

The aim of this study is to verify whether the average value at risk (AVaR) can be a good alternative to the value at risk (VaR) for estimating portfolio losses, especially regarding tail events. To achieve this aim, we use a copula framework to estimate the dependence between the stock returns of a portfolio composed of 94 components of the S&P100 index to compute the AVaR and VaR and compare the results with respect to the Gaussian exponentially weighted moving average (EWMA). To compute the simulated returns, we employ the algorithm used by Biglova et al. (2014) in portfolio selection problems and then backtest the model with Kupiec’s and Christoffersen’s tests. The results are coherent with the literature; in particular, the VaR computed both via the copula and via the EWMA seems to fail to provide an accurate risk measurement while the AVaR with the copula and EWMA appears to be more reliable.

Comparing Australia and the U.S. both prior to and during the Global Financial Crisis (GFC), using a dataset which includes more than six hundred companies, this paper modifies traditional transition matrix credit risk modelling to address two important issues. Firstly, extreme credit risk can have a devastating impact on financial institutions, economies and markets as highlighted by the GFC. It is therefore essential that extreme credit risk is accurately measured and understood. Transition matrix methodology, which measures the probability of a borrower transitioning from one credit rating to another, is traditionally used to measure Value at Risk (VaR), a measure of risk below a specified threshold. An alternate measure to VaR is Conditional Value at Risk (CVaR), which was initially developed in the insurance industry and has been gaining popularity as a measure of extreme market risk. CVaR measures those risks beyond VaR. We incorporate CVaR into transition matrix methodology to measure extreme credit risk. We find significant differences in the VaR and CVaR measurements in both the US and Australian markets, as CVaR captures those extreme risks that are ignored by VaR. We also find a greater differential between VaR and CVaR for the US as compared to Australia, reflecting the more extreme credit risk that was experienced in the US during the GFC. The second issue is that relative industry risk does not stay static over time, as highlighted by the problems experienced by financial sector during the GFC. Traditional transition matrix methodology assumes that all borrowers of the same credit rating transition equally, whereas we incorporate an adjustment based on industry share price fluctuations to allow for unequal transition among industries. The existing CreditPortfolioView model applies industry adjustment factors to credit transition based on macroeconomic variables. The financial sector regulator in Australia, APRA, has found that banks do not favour such credit modelling based on macroeconomic variables due to modelling complexity and forecasting inaccuracy. We use our own iTransition model, which incorporates industry factors derived from equity prices, a much simpler approach than macroeconomic modelling. The iTransition model shows a greater change between Pre-GFC and GFC total credit risk than the traditional model. This means that those industries that were riskiest during the GFC are not the same industries that were riskiest Pre-GFC. The iTransition model also finds that the Australian portfolio, which has a much higher weighting towards financial stocks than the US portfolio, transitions very differently to the more balanced industry-weighted US portfolio. These results highlight the importance of including industry analysis into credit risk modelling. To ensure a thorough analysis of the topic we use various approaches to measuring CVaR. This includes an analytical approach which is based on actual credit ratings as well as a Monte Carlo simulation approach which generates twenty thousand observations for each entity in the data set. We also incorporate historical default probabilities into the model in two different ways, one method using an average historical default rate over time, and the other method using annual default probabilities which vary from year to year. Overall, this comprehensive analysis finds that innovative modelling techniques are better able to account for the impact of extreme risk circumstances and industry composition than traditional transition matrix techniques.

The largest US banks are required by regulatory mandate to estimate the operational risk capital they must hold using an Advanced Measurement Approach (AMA) as defined by the Basel II/III Accords. Most use the Loss Distribution Approach (LDA) which defines the aggregate loss distribution as the convolution of a frequency and a severity distribution representing the number and magnitude of losses, respectively. Estimated capital is a Value-at-Risk (99.9th percentile) estimate of this annual loss distribution. In practice, the severity distribution drives the capital estimate, which is essentially a very high quantile of the estimated severity distribution. Unfortunately, because the relevant severities are heavy-tailed AND the quantiles being estimated are so high, VaR always appears to be a convex function of the severity parameters, causing all widely-used estimators to generate biased capital estimates (apparently) due to Jensen's Inequality. The observed capital inflation is sometimes enormous, even at the unit-of-measure (UoM) level (even billions USD). Herein I present an estimator of capital that essentially eliminates this upward bias. The Reduced-bias Capital Estimator (RCE) is more consistent with the regulatory intent of the LDA framework than implementations that fail to mitigate this bias. RCE also notably increases the precision of the capital estimate and consistently increases its robustness to violations of the i.i.d. data presumption (which are endemic to operational risk loss event data). So with greater capital accuracy, precision, and robustness, RCE lowers capital requirements at both the UoM and enterprise levels, increases capital stability from quarter to quarter, ceteris paribus, and does both while more accurately and precisely reflecting regulatory intent. RCE is straightforward to implement using any major statistical software package.

The aim of the study is to calculate one-day forecasts of the Bayesian value-at-risk (VaR) and expected shortfall (ES) for two kinds of bivariate portfolios and two kinds of datasets. The Bayesian inference for VAR(1)-tCopula-GARCH(1,1), VAR(1)-tBEKK(1,1), and VAR(1)-tDCC(1,1) models and the predictive distribution of ordinary return rates of portfolio are used. The Bayesian VaR and ES fully take into account uncertainty of parameters of model. Moreover, the study also presents the one-day forecasts of VaR with using conditional autoregressive value at risk (CAViaR) with asymmetric slope and ES with employing conditional autoregressive expectiles (CARE) also with asymmetric slope. In order to compare the forecasts of VaR and ES obtained from different models, we use non-Bayesian criteria. The research shows that the calculation of VaR and ES with using tCopula-GARCH model and tBEKK model (or tDCC model for the second dataset) gives similar values of one-day forecasts, taking into account correlation coefficients between predictions from different methods. Moreover the model, which has the highest explanatory power (the highest marginal data density), not in all cases gives the best prediction the VaR and ES considering the non-Bayesian criteria.