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.     

Optimal portfolios with minimum capital requirements

We propose a novel approach to active risk management based on the recent Basel II regulations to obtain optimal portfolios with minimum capital requirements. In order to avoid regulatory penalties due to an excessive number of Value-at-Risk (VaR) violations, capital requirements are minimized subject to a given number of violations over the previous trading year. Capital requirements are based on the recent Basel II amendments to account for the ‘stressed’ VaR, that is, the downside risk of the portfolio under extreme adverse market conditions. An empirical application for two portfolios involving different types of assets and alternative stress scenarios demonstrates that the proposed approach delivers an improved balance between capital requirement levels and the number of VaR exceedances. Furthermore, the risk-adjusted performance of the proposed approach is superior to that of minimum-VaR and minimum-stressed VaR portfolios.     

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.     

Market capitalization and Value-at-Risk

The potential of economic variables for financial risk measurement is an open field for research. This article studies the role of market capitalization in the estimation of Value-at-Risk (VaR). We test the performance of different VaR methodologies for portfolios with different market capitalization. We perform the analysis considering separately financial crisis periods and non-crisis periods. We find that VaR methods perform differently for portfolios with different market capitalization. For portfolios with stocks of different sizes we obtain better VaR estimates when taking market capitalization into account. We also find that it is important to consider crisis and non-crisis periods separately when estimating VaR across different sizes. This study provides evidence that market fundamentals are relevant for risk measurement.     

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.     

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.

     

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.     

Financial crisis,Value-at-Risk forecasts and the puzzle of dependency modeling

Forecasting Value-at-Risk (VaR) for financial portfolios is a crucial task in applied financial risk management. In this paper, we compare VaR forecasts based on different models for return interdependencies: volatility spillover (Engle & Kroner, 1995), dynamic conditional correlations (Engle, 2002, 2009) and (elliptical) copulas (Embrechts et al., 2002). Moreover, competing models for marginal return distributions are applied. In particular, we apply extreme value theory (EVT) models to GARCH-filtered residuals to capture excess returns.Drawing on a sample of daily data covering both calm and turbulent market phases, we analyze portfolios consisting of German Stocks, national indices and FX-rates. VaR forecasts are evaluated using statistical backtesting and Basel II criteria. The extensive empirical application favors the elliptical copula approach combined with extreme value theory (EVT) models for individual returns. 99% VaR forecasts from the EVT-GARCH-copula model clearly outperform estimates from alternative models accounting for dynamic conditional correlations and volatility spillover for all asset classes in times of financial crisis.     

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.     

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.     

A comparison of non-Gaussian VaR estimation and portfolio construction techniques

We propose a multivariate model of returns that accounts for four of the stylised facts of financial data: heavy tails, skew, volatility clustering, and asymmetric dependence with the aim of improving the accuracy of risk estimates and increasing out-of-sample utility of investors’ portfolios. We accommodate volatility clustering, the generalised Pareto distribution to capture heavy tails and skew, and the skewed-t copula to provide for asymmetric dependence. The proposed approach produces more accurate VaR estimates than seven competing approaches across eight data sets encompassing five asset classes. We show that this produces portfolios with higher utility, and lower downside risk than alternative approaches including mean–variance. We confirm that investors can substantially increase utility by accounting for departures from normality.     

Downside risk in Dow Jones Islamic equity indices: Precious metals and portfolio diversification before and after the COVID-19 bear market

Besides great turmoil in financial markets, the COVID-19 pandemic also disrupted the global supply chain, putting the precious metal market into great uncertainty. In this study, we revisit the diversifying role of precious metals – gold, silver, and platinum – for six Dow Jones Islamic (DJI) equity index portfolios using a battery of tests: dynamic conditional correlations (DCCs), four-moment modified value at risk (VaR) and conditional VaR, and global minimum-variance (GMV) portfolio approach. Our empirical results exhibit drastically increased DCCs between sample assets during the COVID period; however, pairing gold with any of the DJI equity indices (except for the Asia-Pacific region) decreases the downside risk of these portfolios. Other precious metals (silver and platinum) do not provide such benefits. Furthermore, we find that a higher allocation of wealth in DJI Japanese equities and gold is required to achieve a GMV portfolio in the post-COVID-19 era, implying higher transaction (hedging) costs to rebalance portfolios (weights) accordingly. Our out-of-sample tests examining the global financial crisis, European debt crisis, and extended sample (2000–2020) periods yield similar findings as gold glitters across all market conditions. Overall, our findings provide notable practical implications for both domestic and international investors.     

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.     

Minimizing CVaR and VaR for a portfolio of derivatives

Value at risk (VaR) and conditional value at risk (CVaR) are frequently used as risk measures in risk management. Compared to VaR, CVaR is attractive since it is a coherent risk measure. We analyze the problem of computing the optimal VaR and CVaR portfolios. We illustrate that VaR and CVaR minimization problems for derivatives portfolios are typically ill-posed. We propose to include cost as an additional preference criterion for the CVaR optimization problem. We demonstrate that, with the addition of a proportional cost, it is possible to compute an optimal CVaR derivative investment portfolio with significantly fewer instruments and comparable CVaR and VaR. A computational method based on a smoothing technique is proposed to solve a simulation based CVaR optimization problem efficiently. Comparison is made with the linear programming approach for solving the simulation based CVaR optimization problem.     

Risk management in the energy markets and Value-at-Risk modelling: a hybrid approach

This paper proposes a set of Value-at-Risk (VaR) models appropriate to capture the dynamics of energy prices and subsequently quantify energy price risk by calculating VaR and expected shortfall measures. Amongst the competing VaR methodologies evaluated in this paper, besides the commonly used benchmark models, a Monte Carlo (MC) simulation approach and a hybrid MC with historical simulation approach, both assuming various processes for the underlying spot prices, are also being employed. All VaR models are empirically tested on eight spot energy commodities that trade futures contracts on the New York Mercantile Exchange (NYMEX) and the constructed Spot Energy Index. A two-stage evaluation and selection process is applied, combining statistical and economic measures, to choose amongst the competing VaR models. Finally, both long and short trading positions are considered as it is of utmost importance for energy traders and risk managers to be able to capture efficiently the characteristics of both tails of the distributions.     

Portfolio Diversification Effects of Downside Risk

Risk managers use portfolios to diversify away the unpricedrisk of individual securities. In this article we compare thebenefits of portfolio diversification for downside risk in casereturns are normally distributed with the case of fat-taileddistributed returns. The downside risk of a security is decomposedinto a part which is attributable to the market risk, an idiosyncraticpart, and a second independent factor. We show that the fat-tailed-baseddownside risk, measured as value-at-risk (VaR), should declinemore rapidly than the normal-based VaR. This result is confirmedempirically.     

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.     

Backtesting Derivative Portfolios with Filtered Historical Simulation (FHS)

Filtered historical simulation provides the general framework to our backtests of portfolios of derivative securities held by a large sample of financial institutions. We allow for stochastic volatility and exchange rates. Correlations are preserved implicitly by our simulation procedure. Options are repriced at each node. Overall results support the adequacy of our framework, but our VaR numbers are too high for swap portfolios at long horizons and too low for options and futures portfolios at short horizons.     

Ethical and unethical investments under extreme market conditions

This study investigates the time-varying volatility and risk measures of ethical and unethical investments. We apply the Bayesian Markov-switching generalized autoregressive conditional heteroscedasticity (MS-GARCH) approach to compute the value-at-risk (VaR) and expected shortfall (ES) of ethical and unethical indices returns, which allows for detecting the differences between ethical and unethical investments. The innovative finding of our study is that ethical investments are less affected during global financial crises compared with unethical and conventional investments. The policy implication of this study is that investors should consider ethical investments as a hedging asset for their portfolios during extreme market conditions.