Portfolio credit-risk optimization

This paper evaluates several alternative formulations for minimizing the credit risk of a portfolio of financial contracts with different counterparties. Credit risk optimization is challenging because the portfolio loss distribution is typically unavailable in closed form. This makes it difficult to accurately compute Value-at-Risk (VaR) and expected shortfall (ES) at the extreme quantiles that are of practical interest to financial institutions. Our formulations all exploit the conditional independence of counterparties under a structural credit risk model. We consider various approximations to the conditional portfolio loss distribution and formulate VaR and ES minimization problems for each case. We use two realistic credit portfolios to assess the in- and out-of-sample performance for the resulting VaR- and ES-optimized portfolios, as well as for those which we obtain by minimizing the variance or the second moment of the portfolio losses. We find that a Normal approximation to the conditional loss distribution performs best from a practical standpoint.     

Applying VaR to REITs: A comparison of alternative methods

This study employs five methods to calculate the VaR of twelve REITs portfolios and evaluates the accuracy of these methods. Firstly, we find that the VaR varies among individual portfolios. The Hotel REITs has consistently the largest VaR. The low-leveraging portfolio tends to have the largest VaR measured by the parametric methods, while the high leveraging portfolio has the largest VaR calculated by the non-parametric methods. Secondly, each method performs differently at different confidence levels, and no method dominates the others. At the 95% confidence level, the EWMA method performs relatively well. The EQWMA and the two non-parametric methods perform equivalently and slightly overestimate VaRs. The EQWMA_T method ranks the bottom and significantly overestimates VaRs for all portfolios. At the 99% confidence level, the EQWMA method performs the best. The EQWMA_T and the two non-parametric methods perform equivalently and may overestimate VaR for all portfolios. The EWMA method turns out to be the worst and tends to underestimate the VaR. These findings may provide more insights for institutional real estate investors.     

On Bayesian Value at Risk: From Linear to Non-Linear Portfolios

This paper proposes the use of Bayesian approach to implement Value at Risk (VaR) model for both linear and non-linear portfolios. The Bayesian approach provides risk traders with the flexibility of adjusting their VaR models according to their subjective views. First, we deal with the case of linear portfolios. By imposing the conjugate-prior assumptions, a closed-form expression for the Bayesian VaR is obtained. The Bayesian VaR model can also be adjusted in order to deal with the ageing effect of the past data. By adopting Gerber-Shiu's option-pricing model, our Bayesian VaR model can also be applied to deal with non-linear portfolios of derivatives. We obtain an exact formula for the Bayesian VaR in the case of a single European call option. We adopt the method of back-testing to compare the non-adjusted and adjusted Bayesian VaR models with their corresponding classical counterparts in both linear and non-linear cases.     

Mean–variance portfolio selection with ‘at-risk’ constraints and discrete distributions

We examine the impact of adding either a VaR or a CVaR constraint to the mean–variance model when security returns are assumed to have a discrete distribution with finitely many jump points. Three main results are obtained. First, portfolios on the VaR-constrained boundary exhibit (K + 2)-fund separation, where K is the number of states for which the portfolios suffer losses equal to the VaR bound. Second, portfolios on the CVaR-constrained boundary exhibit (K + 3)-fund separation, where K is the number of states for which the portfolios suffer losses equal to their VaRs. Third, an example illustrates that while the VaR of the CVaR-constrained optimal portfolio is close to that of the VaR-constrained optimal portfolio, the CVaR of the former is notably smaller than that of the latter. This result suggests that a CVaR constraint is more effective than a VaR constraint to curtail large losses in the mean–variance model.     

Bank Capital Regulation of Trading Portfolios: An Assessment of the Basel Framework

In setting minimum capital requirements for trading portfolios, the Basel Committee on Banking Supervision (1996, 2011a, 2013) initially used Value‐at‐Risk (VaR), then both VaR and stressed VaR (SVaR), and most recently, stressed Conditional VaR (SCVaR). Accordingly, we examine the use of SCVaR to measure risk and set these requirements. Assuming elliptically distributed asset returns, we show that portfolios on the mean‐SCVaR frontier generally lie away from the mean‐variance (M‐V) frontier. In a plausible numerical example, we find that such portfolios tend to have considerably higher ratios of risk (measured by, e.g., standard deviation) to minimum capital requirement than those of portfolios on the M‐V frontier. Also, we find that requirements based on SCVaR are smaller than those based on both VaR and SVaR but exceed those based on just VaR. Finally, we find that requirements based on SCVaR are less procyclical than those based on either VaR or both VaR and SVaR. Overall, our paper suggests that the use of SCVaR to measure risk and set requirements is not a panacea.     

On the performance of the minimum VaR portfolio

Alexander and Baptista [2002. Economic implications of using a mean-value-at-risk (VaR) model for portfolio selection: A comparison with mean–variance analysis. Journal of Economic Dynamics and Control 26: 1159–93] develop the concept of mean-VaR efficiency for portfolios and demonstrate its very close connection with mean–variance efficiency. In particular, they identify the minimum VaR portfolio as a special type of mean–variance efficient portfolio. Our empirical analysis finds that, for commonly used VaR breach probabilities, minimum VaR portfolios yield ex post returns that conform well with the specified VaR breach probabilities and with return/risk expectations. These results provide a considerable extension of evidence supporting the empirical validity and tractability of the mean-VaR efficiency concept.     

Accrual mispricing,value-at-risk,and expected stock returns

We investigate the extent to which a parsimonious measure of maximum likely loss that captures the tail risk of returns—known as value-at-risk (VaR)—explains the relationship between accruals and the cross-sectional dispersion of expected stock returns. We construct portfolios based on Sloan’s (Account Rev 71(3):289–315, 1996) total accruals (TA) measure and individual asset-level VaR, which reflects the dynamic behavior of the asset distribution. We document that VaR is in congruence with portfolio-level accruals and that there is a significant positive relationship between VaR and the cross-section of portfolio returns. Allowing a double-sort involving VaR and TA further suggests that the spread between low- and high-TA portfolios is significantly attenuated after controlling for VaR. We also conduct a firm-level cross-sectional regression analysis and demonstrate that the TA- and VaR-based characteristics—but not the factor-mimicking portfolios—are compensated with higher expected returns, and that VaR neither subsumes nor is subsumed by TA. Finally, our cross-sectional decomposition analysis suggests that the firm-level VaR captures at least 7% of the accrual premium even in the presence of size and book-to-market. These findings lend support for the mispricing explanation of the accrual anomaly.

     

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.     

The Risk Map: A new tool for validating risk models

This paper presents a new method to validate risk models: the Risk Map. This method jointly accounts for the number and the magnitude of extreme losses and graphically summarizes all information about the performance of a risk model. It relies on the concept of a super exception, which is defined as a situation in which the loss exceeds both the standard Value-at-Risk (VaR) and a VaR defined at an extremely low probability. We then formally test whether the sequences of exceptions and super exceptions are rejected by standard model validation tests. We show that the Risk Map can be used to validate market, credit, operational, or systemic risk estimates (VaR, stressed VaR, expected shortfall, and CoVaR) or to assess the performance of the margin system of a clearing house.     

Credit risk in the leasing industry

This paper is devoted to the credit risk modeling issues of retail lease portfolios. Using a re-sampling method, I estimate the probability density function of losses and VaR measures in a portfolio of 46,732 leases issued between 1990 and 2000 by a major European financial institution. My results show that physical collaterals play a major role in reducing the credit risk associated with lease portfolios. However, because of insufficient recognition of such collaterals under the new regulatory capital framework (Basel II), significant differences are observed between the estimated capital requirements and those calculated in accordance with the various Basel II approaches.     

Dynamic factor Value-at-Risk for large heteroskedastic portfolios

We propose a methodology that can efficiently measure the Value-at-Risk (VaR) of large portfolios with time-varying volatility and correlations by bringing together the established historical simulation framework and recent contributions to the dynamic factor models literature. We find that the proposed methodology performs well relative to widely used VaR methodologies, and is a significant improvement from a computational point of view.     

Value-at-Risk: a multivariate switching regime approach

This paper analyses the application of a switching volatility model to forecast the distribution of returns and to estimate the Value-at-Risk (VaR) of both single assets and portfolios. We calculate the VaR value for 10 Italian stocks and a number of portfolios based on these stocks. The calculated VaR values are also compared with the variance–covariance approach used by JP Morgan in RiskMetrics™ and GARCH(1,1) models. Under backtesting, the VaR values calculated using the switching regime beta model are preferred to both other methods. The Proportion of Failure and Time Until First Failure tests [The Journal of Derivatives (1995) 73–84] confirm this result.     

VaR和Z′评分模型在农企信用担保风险管理中的应用

对农村企业信用担保风险的有效管理是农村企业信用担保机构存在的基础,因此,农村企业信用担保机构应建立能够贯穿整个担保项目始终的信用担保风险管理体系。而选用能兼顾事前风险预测和事后风险监控的VaR模型和Z′评分模型作为农企信用担保机构定量风险管理的手段,可以有效实现其风险管理目标。     

Evaluating portfolio Value-at-Risk using semi-parametric GARCH models

In this paper we examine the usefulness of multivariate semi-parametric GARCH models for evaluating the Value-at-Risk (VaR) of a portfolio with arbitrary weights. We specify and estimate several alternative multivariate GARCH models for daily returns on the S&P 500 and Nasdaq indexes. Examining the within-sample VaRs of a set of given portfolios shows that the semi-parametric model performs uniformly well, while parametric models in several cases have unacceptable failure rates. Interestingly, distributional assumptions appear to have a much larger impact on the performance of the VaR estimates than the particular parametric specification chosen for the GARCH equations.     

How to choose the return model for market risk? Getting towards a right magnitude of stressed VaR

Value at Risk (VaR) and stressed value at Risk (SVaR) or expected shortfall are important risk measures widely used in the financial services industry for risk management and market risk capital computation. Fundamental to any (S)VaR model is the choice of the return type model for each risk factor. Because the resulting SVaR numbers are highly sensitive to the chosen return type model it is important to make a prudent choice on the return type modelling. We propose to estimate the return type model from historic data without making an a priori model assumption on the return model. We explain the fundamentals of return type modelling and how it impacts the magnitude of SVaR. We further show how to obtain a global return type model from a set of similar return type models by using geometric calculus. Numerical simulations and illustrations are provided. In this paper, we consider interest rate data, but the proposed methodology is general and can be applied to any other asset class such as inflation, credit spread, equity or fx.     

Saddlepoint approximation of CreditRisk+

CreditRisk⁺ is an influential and widely implemented model of portfolio credit risk. As a close variant of models long used for insurance risk, it retains the analytical tractability for which the insurance models were designed. Value-at-risk (VaR) can be obtained via a recurrence-rule algorithm, so Monte Carlo simulation can be avoided. Little recognized, however, is that the algorithm is fragile. Under empirically realistic conditions, numerical error can accumulate in the execution of the recurrence rule and produce wildly inaccurate results for VaR.This paper provides new tools for users of CreditRisk⁺ based on the cumulant generating function (cgf) of the portfolio loss distribution. Direct solution for the moments of the loss distribution from the cgf is almost instantaneous and is computationally robust. Thus, the moments provide a convenient, quick and independent diagnostic on the implementation and execution of the standard solution algorithm. Better still, with the cgf in hand we have an alternative to the standard algorithm. I show how tail percentiles of the loss distribution can be calculated quickly and easily by saddlepoint approximation. On a large and varied sample of simulated test portfolios, I find a natural complementarity between the two algorithms: Saddlepoint approximation is accurate and robust in those situations for which the standard algorithm performs least well, and is less accurate in those situations for which the standard algorithm is fast and reliable.     

Transitional credit modelling and its relationship to market value at risk: an Australian sectoral perspective

Internal credit risk modelling is important for banks for the calculation of capital adequacy in terms of the Basel Accords, and for the management of sectoral exposure. We examine Credit Value at Risk (VaR), Conditional Credit Value at Risk (Credit CVaR) and the relationship between market and credit risk. Significant association is found between different Credit CVaR methods, and between market and credit risk. Simpler Credit CVaR methods are found to be viable alternatives to more complex methodology. The relationship between market and credit risk is used to develop a new model that allows banks to incorporate industry risk into transition modelling, without macroeconomic analysis.     

Credit Risk Versus Capital Requirements under Basel II: Are SME Loans and Retail Credit Really Different?

Under Basel II, retail and SME credit (R&SME) receive special treatment because of a supposedly smaller exposure to systemic risk. Most research on this issue has been based on parameterized credit risk models. We present new evidence by applying Carey's (Carey, Mark. “Credit Risk in Private Debt Portfolios.” Journal of Finance 53, no. 4 (1998), 1363–1387.) nonparametric Monte-Carlo resampling method to two banks' complete loan portfolios. By exploiting that a sub-sample of all borrowers has been assigned an internal rating by both banks, we can compare the credit loss distributions for the three credit types, and compute both economic and regulatory capital under Basel II. We also test if our conclusions are sensitive to the definitions of R&SME credit. Our findings show that R&SME portfolios are usually riskier than corporate credit. Special treatment under Basel II is thus not justified. JEL classification: C14, C15, G21, G28, G33.     

Scenario Simulation: Theory and methodology

This paper presents a new simulation methodology for quantitative risk analysis of large multi-currency portfolios. The model discretizes the multivariate distribution of market variables into a limited number of scenarios. This results in a high degree of computational efficiency when there are many sources of risk and numerical accuracy dictates a large Monte Carlo sample. Both market and credit risk are incorporated. The model has broad applications in financial risk management, including value at risk. Numerical examples are provided to illustrate some of its practical applications.     

论银行市场风险的资本计提——兼评内部模型法的适用性

市场风险及其监管资本要求的计量历来为业界和监管当局所关注。近期,次贷危机爆发导致的市场动荡使得全球银行业和监管当局开始重新审视其市场风险管理和监管资本要求。文章结合国际银行业和监管机构计量市场风险及其监管资本要求的当前做法,针对我国银行业的实际情况,重点探索了内部模型法在我国银行业的适用性,尤其是从方法论、特殊风险计量、验证等角度探讨了内部模型法的主要工具——风险价值体系在我国银行业计量市场风险及其监管资本要求的适用性,并从方法论和应用层面提出了相应的政策建议。