Coherent risk measures under filtered historical simulation

Recent studies have strongly criticised conventional VaR models for not providing a coherent risk measure. Acerbi provides the intuition for an entire family of coherent measures of risk known as “spectral risk measures” [Spectral measures of risk: A coherent representation of subjective risk aversion. Journal of Banking and Finance 26 (7) (2002) 1505–1518]. In this study we illustrate how the Filtered Historical Simulation [Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t look back. Risk 11, 100–104; Barone-Adesi, Giannopoulos, K., Vosper, L., 1999. VaR without correlations for non-linear portfolios. Journal of Futures Markets 19, 583–602], can provide an improved methodology for calculating the Expected Shortfall. Thereafter, we prove that these new risk measures are spectral and are coherent as well, following Acerbi. Furthermore, we provide the statistical error formula that allows to calculate the error for our model.     

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.     

Forecasting market risk of portfolios: copula-Markov switching multifractal approach

This paper proposes a new methodology for modeling and forecasting market risks of portfolios. It is based on a combination of copula functions and Markov switching multifractal (MSM) processes. We assess the performance of the copula-MSM model by computing the value at risk of a portfolio composed of the NASDAQ composite index and the S&P 500. Using the likelihood ratio (LR) test by Christoffersen [1998. “Evaluating Interval Forecasts.” International Economic Review 39: 841–862], the GMM duration-based test by Candelon et al. [2011. “Backtesting Value at Risk: A GMM Duration-based Test.” Journal of Financial Econometrics 9: 314–343] and the superior predictive ability (SPA) test by Hansen [2005. “A Test for Superior Predictive Ability.” Journal of Business and Economic Statistics 23, 365–380] we evaluate the predictive ability of the copula-MSM model and compare it to other common approaches such as historical simulation, variance–covariance, RiskMetrics, copula-GARCH and constant conditional correlation GARCH (CCC-GARCH) models. We find that the copula-MSM model is more robust, provides the best fit and outperforms the other models in terms of forecasting accuracy and VaR prediction.     

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.     

Nonparametric,conditional pricing of higher order multivariate contingent claims

This paper describes and applies a nonparametric model for pricing multivariate contingent claims. Multivariate contingent claims are contracts whose payoffs depend on the future prices of more than one underlying variable. The pricing however of these kinds of contracts represents a challenge. All known models are adaptations of earlier ones that have been introduced to price plain vanilla calls and puts. They are imposing strong assumptions on the distributional properties of the underlying variables. In contrast, this study adopts a methodology that relaxes such restrictions. Following [Barone-Adesi, G., Bourgoin, F., Giannopoulos, K., 1998. Don’t Look Back, Risk 11 (August), 100–104; Barone-Adesi, G., Engle, R., Mancini, L., 2004. GARCH Options in Incomplete Markets, mimeo, University of Applied Sciences of Southern Switzerland; Long, X., 2004. Semiparametric Multivariate GARCH Model, mimeo, University of California, Riverside], multivariate pathways for a set of underlying variables are constructed before the option payoffs are computed. This enables the covariances, in addition to the means and variances, to be modelled in a dynamic and nonparametric manner. The model is particular suitable for options whose payoffs depend on variables that are characterised by high nonlinearities and extremes and on higher order multivariate options whose underlying variables are more unlikely to conform to a common theoretical distribution.     

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.     

How does the choice of Value-at-Risk estimator influence asset allocation decisions?

Considering the growing need for managing financial risk, Value-at-Risk (VaR) prediction and portfolio optimisation with a focus on VaR have taken up an important role in banking and finance. Motivated by recent results showing that the choice of VaR estimator does not crucially influence decision-making in certain practical applications (e.g. in investment rankings), this study analyses the important question of how asset allocation decisions are affected when alternative VaR estimation methodologies are used. Focusing on the most popular, successful and conceptually different conditional VaR estimation techniques (i.e. historical simulation, peak over threshold method and quantile regression) and the flexible portfolio model of Campbell et al. [J. Banking Finance. 2001, 25(9), 1789–1804], we show in an empirical example and in a simulation study that these methods tend to deliver similar asset weights. In other words, optimal portfolio allocations appear to be not very sensitive to the choice of VaR estimator. This finding, which is robust in a variety of distributional environments and pre-whitening settings, supports the notion that, depending on the specific application, simple standard methods (i.e. historical simulation) used by many commercial banks do not necessarily have to be replaced by more complex approaches (based on, e.g. extreme value theory).     

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.     

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.     

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.     

Value-at-risk in a market subject to regime switching

Many empirical researches report that value-at-risk (VaR) measures understate the actual 1% quantile, while for Inui, K., Kijima, M. and Kitano, A., VaR is subject to a significant positive bias. Stat. Probab. Lett., 2005, 72, 299–311. proved that VaR measures overstate significantly when historical simulation VaR is applied to fat-tail distributions. This paper resolves the puzzle by developing a regime switching model to estimate portfolio VaR. It is shown that our model is able to correct the underestimation problem of risk.     

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.     

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.     

Intertemporal asset allocation: A comparison of methods

This paper compares two recent Monte Carlo methods advocated for the computation of optimal portfolio rules. The candidate methods are the approach based on Monte Carlo with Malliavin Derivatives (MCMD) proposed by Detemple, Garcia and Rindisbacher [Detemple et al., 2003. A Monte-Carlo method for optimal portfolios. Journal of Finance 58, 401–406] and the approach based on Monte Carlo with regression (MCR) of Brandt, Goyal, Santa-Clara and Stroud [Brandt et al., 2003. A simulation approach to dynamic portfolio choice with an application to learning about return predictability. Working paper, Wharton School]. Our comparisons are carried out in the context of various intertemporal portfolio choice problems with two assets, a risky asset and a riskless asset, and different configurations of the state variables. The specifications studied include a linear model with a single state variable admitting an exact solution and a non-linear model with two state variables that requires a purely numerical resolution. The accuracies of the candidate methods are compared. We provide, in particular, efficiency plots displaying the speed–accuracy trade-off for various selections of the relevant simulation and discretization parameters. MCMD is shown to dominate in all the settings considered.     

Improving expected tail loss estimates with neural networks

Expected tail loss (ETL) and other ‘coherent’ risk measures are rapidly gaining acceptance amongst risk managers due to the limitations of value‐at‐risk (VaR) as a risk measure. In this article we explore the use of multilayer perceptron supervised neural networks to improve our estimates of ETL numbers using information from both tails of the distribution. We compare the results with the historical simulation approach to the estimation of VaR and ETL. The evaluation results indicate that the ETL estimates using neural networks are superior to historical simulation ETL estimates in all periods except for one, and in that case the historical ETL is slightly superior. Overall, therefore, when the whole period is considered, our results indicate that the network estimates of ETL are superior to the historical ones. Finally, one of the most interesting results of the study is the fact that the neural networks seem to indicate that VaR and ETL (as a function of VaR itself) are dependent not only on the negative returns observed, but also on large positive returns, which indicates that too much emphasis on losses could lead us to overlook important risk information arising from large positive returns. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

Portfolio optimization under Expected Shortfall: contour maps of estimation error

The contour maps of the error of historical and parametric estimates of the global minimum risk for large random portfolios optimized under the Expected Shortfall (ES) risk measure are constructed. Similar maps for the VaR of the ES-optimized portfolio are also presented, along with results for the distribution of portfolio weights over the random samples and for the out-of-sample and in-sample estimates for ES. The contour maps allow one to quantitatively determine the sample size (the length of the time series) required by the optimization for a given number of different assets in the portfolio, at a given confidence level and a given level of relative estimation error. The necessary sample sizes invariably turn out to be unrealistically large for any reasonable choice of the number of assets and the confidence level. These results are obtained via analytical calculations based on methods borrowed from the statistical physics of random systems, supported by numerical simulations.     

Evaluating the impact of market reforms on Value-at-Risk forecasts of Chinese A and B shares

This paper analyses the time-varying conditional correlations between Chinese A and B share returns using the Dynamic Conditional Correlation (DCC) model of Engle [Engle, R.F. (2002), “Dynamic Conditional Correlation: A Simple Class of Multivariate Generalized Autoregressive Conditional Heteroskedasticity Models”, Journal of Business and Economic Statistics, 20, 339–350.]. The results show that the conditional correlations increased substantially following the B share market reform, whereby Chinese investors were permitted to purchase B shares. However, this increase in correlations was found to have begun well before the B share market reform. This result has significant implication relating to the structure of the information flow between the markets for the two classes of shares. Value-at-Risk (VaR) threshold forecasts are used to analyse the importance of accommodating dynamic conditional correlations between Chinese A and B shares, and thus reflects the impact of the changes in information flow on the risk evaluation of a diversified portfolio. The competing VaR forecasts are analysed using the Unconditional Coverage, Serial Independence and Conditional Coverage tests of Christoffersen [Christoffersen (1998), “Evaluating Interval Forecasts”, International Economic Review, 39, 841–862], and the Time Until First Failure Test of Kupiec [Kupiec, P.H., (1995), “Techniques for Verifying the Accuracy of Risk Measurements Models”, Journal of Derivatives, 73–84]. The results offer mild support for the DCC model over its constant conditional correlation counterpart.     

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.     

Modeling fund and portfolio risk: A bi-modal approach to analyzing risk in turbulent markets

Following the financial crisis of 2008, it has been argued that Value at Risk (VaR), and risk analysis in general, failed to alert risk managers of the turbulence on the horizon. This is a misguided view that should not have come as a surprise because many widely circulated academic papers and discussions suggested, well before the crisis, that simple VaR results could easily be misinterpreted if the circumstances for its proper use are not fully understood. This paper addresses some ways in which VaR concepts may be applied more effectively. Non-standard Monte Carlo simulations are utilized. Whereas standard mean–variance defined methodologies using Monte Carlo analysis may not capture how “fat” a lower tail may actually be, a bi-modal switching structure between assumed normal periods and possible turbulent economic periods may help resolve the problem. Lower boundaries (worst case paths) of the different (normal versus bi-modal) processes are mapped to illustrate implied riskiness of portfolios if turbulence occurs. The analysis implies that no mechanical risk analysis is sufficiently divorced from a judgment call about possible market disruptions; however, a bi-modal approach allows quantification of the said judgment in conjunction with empirical observations from history.