VaR Measures Research Articles

An Econometric Approach to Incorporating Non-Normality in VaR Measurement

Following the recent financial crises, there has been a proliferation of new risk management and portfolio construction approaches. These approaches all endeavour to better quantify and manage risk by accounting for the stylised facts of financial time series mainly heavy and skewed tails, volatility clustering and converging correlations. Capturing all these stylised facts in a coherent framework has proved to be an elusive and knotty task. We here propose a pure econometric framework that captures all the stylised facts satisfactorily. We use three data sets to show how the approach is implemented in VaR forecasting and correlation analysis. We show how an investment portfolio can be constructed in order to optimise reserve capital holding. The approach employed is linear programming (LP) computable, satisfies second degree stochastic dominance and outperforms the general mean/VaR quadratic optimisation to arrive at efficient asset allocation.

PERFORMANCE OF CONDITIONAL MODELS IN GOLD RISK MANAGEMENT

ABSTRACT Even with studies to confront different risk models for gold, there is no consensus about what is the best approach or models when considering the presence of extreme negative values. To that, we employ a backtesting in conditional models with distinct distributions in order to estimate VaR and ES risk measures and, thus, find a pattern for the risk of investments in gold. We verify that the EVT approach has more conservative and volatile risk estimates, with satisfactory results in extreme situations.

Correlations and risk contagion between mixed assets and mixed-asset portfolio VaR measurements in a dynamic view: An application based on time varying copula models

In this paper, we apply time varying Gaussian and SJC copula models to study the correlations and risk contagion between mixed assets: financial (stock), real estate and commodity (gold) assets in China firstly. Then we study the dynamic mixed-asset portfolio risk through VaR measurement based on the correlations computed by the time varying copulas. This dynamic VaR-copula measurement analysis has never been used on mixed-asset portfolios. The results show the time varying estimations fit much better than the static models, not only for the correlations and risk contagion based on time varying copulas, but also for the VaR-copula measurement. The time varying VaR-SJC copula models are more accurate than VaR-Gaussian copula models when measuring more risky portfolios with higher confidence levels. The major findings suggest that real estate and gold play a role on portfolio risk diversification and there exist risk contagion and flight to quality between mixed-assets when extreme cases happen, but if we take different mixed-asset portfolio strategies with the varying of time and environment, the portfolio risk will be reduced.

Historical VaR, CVaR and ES to Liquidation: 'A New Tool to Measure Risk by Regulators and Fund Managers'

Historical VaR, CVaR and ES (Expected Shortfall) to LIQUIDATION Software is a model characterized by its straightforwardness, allowing regulators measure risk using a standard database of primitive factors and portfolio positions only, leaving little error margin in comparing market risk for different financial funds. As such, it should be a tool of preference for government and supra-government regulatory institutions and is the required VaR measure of government regulators of private pension funds in Latin America. The proposed model is based in the fundamental premise that there is a limit to the amount a financial asset can be sold in a given trading day without the market detecting the fund is in unwind mode. To remain consistent with my previous models, I am not shy of dispensing with premises behind valuation models when the real world demands so, like short-term Efficient-Market-Hypothesis, EMH. In addition, the model includes a new measure of risk: a liquidity haircut for each asset and trade size. Like in my previous work, there is no Horizon H – due to different asset liquidity – in calculating the risk metrics. The algorithm developed computes simultaneously VaR and ES. (Since CVaR is defined in the specialized literature as equal to ES, I shall dispense of the former, making sole reference to ES throughout the paper.) In each simulation run Individual and Component VaR are calculated together with Historical VaR, with a good deal of information obtained in a relatively short period of time. The following version, to be release shortly, will include Individual and Component ES, as well as Marginal and Incremental VaR and ES.

Different Risk Adjusted Performance Measures for Equity Mutual Funds: A Comparative Study of VAR and Traditional Measures

Traditional measures of performance evaluation are in vogue since long, however, Value at Risk (VaR) approaches are making their place in portfolio management industry from the last ten years. Value at Risk (VaR) approach focuses on the downside volatility of portfolio, thus making the investor clear about the maximum possible loss on his portfolio. In today‘s highly volatile environment, an investor is concerned more about the downside risk rather upward swing in the portfolio. He would hesitate to invest in a portfolio having more downward risk. Present paper makes an attempt to compare both the traditional and VaR performance measures and explore whether differences exist in ranking of funds using both the approaches. Results indicated that Sharpe ratio and Normal VaR results are same when used independently, while there is difference in results using Treynor and Jensen‘s Alpha.

Economics of eBay’s buyer protection plan

This paper explores the financial viability of eBay’s Buyer Protection Plan. We explore their warranty model assuming exponential auction listing and claims times, under competing assumptions of Normal (classical portfolio theory) versus Paretian claims (industry practice) distributions using a normative algorithm. EBay’s model viability was analyzed under three risk metrics – mean-variance risk of classic portfolio theory; value at risk used in regulation such as the Basel Accords; and tail value at risk which is preferred by academics. Five main findings of the research are: (1) long-term financial viability is set by boundary conditions that can be managed in eBay’s contract terms, (2) commonly used but simplistic assumptions of Gaussian distributions of contract failures can be misleading, and keeping the Buyer Protection Plan viable demands more realistic distributions, (3) value at risk measures provide more information about viability than expectations, (4) tail value at risk measures an lead to sudden and extreme variations in fund value over the parameter range and flawed contracting decisions, and (5) tail value at risk is the preferable risk measure for assessing Buyer Protection Plan viability, but it needs to be used with an assumption of Lomax contract failures to support profitable contracting. Our analysis showed substantial improvements in claims management when tail value at risk measures were used instead of conventional VaR measures. This implies that current strategies, and prices for coverage may result in losses to eBay, in current practice.

Risk management with expectiles

Expectiles (EVaR) are a one-parameter family of coherent risk measures that have been recently suggested as an alternative to quantiles (VaR) and to expected shortfall (ES). In this work we review their known properties, we discuss their financial meaning, we compare them with VaR and ES and we study their asymptotic behaviour, refining some of the results in Bellini et al. [(2014). “Generalized Quantiles as Risk Measures.” Insurance: Mathematics and Economics, 54:41–48]. Moreover, we present a real-data example for the computation of expectiles by means of simple Garch(1,1) models and we assess the accuracy of the forecasts by means of a consistent loss function as suggested by Gneiting [(2011). “Making and Evaluating Point Forecast.” Journal of the American Statistical Association, 106 (494): 746–762]. Theoretical and numerical results indicate that expectiles are perfectly reasonable alternatives to VaR and ES risk measures.

Estimation of Value-at-Risk with Minimal Specification Error

In this paper we provide a new statistical approach to estimate VaR with minimal specification error. In our approach we use current market information obtained from traded options to infer risk-neutral moments of the underlying asset returns over different horizons. For each horizon, the moments are fitted to a general 4-moment Variance Gamma risk-neutral probability distribution of future market prices. We also compute a Radon-Nikodym derivative and its probability distribution, and use this to transform the risk-neutral distribution of the underlying to its empirical distribution. The empirical distribution is then used to find the VaR. The novelty of our approach is that it avoids unnecessary mis-specification bias as we do not need to assume a particular empirical distribution of the underlying asset returns. As option prices trade frequently, the risk-neutral distribution can be updated frequently for transformation into the empirical distribution. This provides for timely updating of the VaR measures. The fitted risk-neutral Variance Gamma distribution is popular in finance due to its ability to model heavy tails and skewness. Our key contributions are in providing an alternative new approach to modeling VaR, and also in showing that underestimation of risk is largely not due to VaR itself but perhaps due to mis-specification errors which we minimize in our approach. We show that our method of measuring VaR clearly captures large tail risk in the empirical examples on S&P 500 index.

Study on Dynamic Risk Measurement Based on ARMA-GJR-AL Model

This paper established the ARMA-GJR-AL model of dynamic risk VaR and CVaR measurement. Considering from aspects of the correlation and volatility and residual distribution characteristics, studying the dynamic risk measures of VaR and CVaR based on ARMA-GJR-AL model. Through empirical research, Risk prediction and accuracy of inspection are given of the Shanghai stock market and the New York stock market. And we study the effectiveness of the model. The results show that the dynamic risk measurement model based on AL distribution is more reasonable and applicability, so it can effectively measure risk.

Predicting Conditional Autoregressive Value-at-Risk for Stock Markets during Tranquil and Turbulent Periods

This paper analyzes the predictive performance of the Conditional Autoregressive Value at Risk (CAViaR) developed by Engle & Manganelli (2004) for major equity markets during tranquil and turbulent periods. The CAViaR model shifts the focus of attention from the distribution of returns directly to the behaviour of the quantile. We compare the predictive performance of four alternative CAViaR specifications, namely Adaptive, Symmetric Absolute Value, Asymmetric Slope and Indirect GARCH(1,1) models due to Engle & Manganelli (2004) along with the improved asymmetric CAViaR (I-CAViaR) model due to Huang et al. (2009). We employ daily returns for six stock markets indices, namely S & P500, FTSE100, NIKKEI225, DAX30, CAC40 and Athens Exchange General index for the period January 2, 1995 to August 23, 2013. We compare the predictive performance of the alternative specifications for three subperiods: before, during and after the recent 2007-2009 financial crisis. The comparison is done with the use of a battery of tests which includes unconditional and conditional coverage tests, the Dynamic Quantile high-order independence test and the White (2000) empirical coverage probability and predictive quantile loss tests. The main findings of the present analysis is that the CAViaR quantile regression models and the I-CAViaR model have shown significant success in predicting the VaR measure for various periods although this performance varies over the three periods before, during and after the 2007-2009 financial crisis.

Risk Measuring Under Liquidity Risk

We present a general framework for measuring the liquidity risk. The theoretical framework de fines risk measures that incorporate the liquidity risk into the standard risk measures. We consider a one-period risk measurement model. The liquidity risk is defi ned as the risk that a given security or a portfolio of securities cannot be easily sold or bought by the financial institutions without causing signi ficant changes in prices. The new risk measures are decomposed into two terms, one measuring the risk of the future value of a given position in a security or a portfolio of securities, and the other the initial cost of this position. Within the framework of coherent risk measures, the risk measures applied to the random part of the future value of a position in a determinatesecurity present some diff erences with respect to the standard riskmeasures. In particular, they are increasing monotonic and convex cash subadditive on long positions. The contrary, in certain situations, holds for the sell positions. For the long positions case, we provide these new risk measures with a dual representation. In some speci fic cases also the sell positions can beequipped with a dual representation. By using convex risk measures, we apply our framework to the situation in which financial institutions break up large trades into many small ones. Dual representation results are also obtained for both positions in securities and portfolios. We give many practical examples of risk measures and derive for each of them the respective capital requirement, and show that our setting can also be applied to contingent securities. As aparticular example, we discuss the VaR measure.

Optimal Reinsurance Under VaR and CTE Risk Measures When Ceded Loss Function is Concave

Cai et al. (2008) explored the optimal reinsurance designs among the class of increasing convex reinsurance treaties under VaR and CTE risk measures. However, reinsurance contracts always involve a limit on the ceded loss function in practice, and thus it may not be enough to confine the analysis to the class of convex functions only. The object of this article is to present an optimal reinsurance policy under VaR and CTE optimization criteria when the ceded loss function is in a class of increasing concave functions and the reinsurance premium is determined by the expected value principle. The outcomes reveal that the optimal form and amount of reinsurance depend on the confidence level p for the risk measure and the safety loading θ for the reinsurance premium. It is shown that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is optima, while the full reinsurance with a policy limit is optima under CTE optimization criterion. Some illustrative examples are provided.

Histogram-valued data on value at risk measures: a symbolic approach for risk attribution

In this article, we develop the concept of histogram-valued data on value at risk for the classification of hedge fund risk. By using recent developments in data mining, it is a question of the classification of heterogeneous data in order to sort hedge funds by risk class. In practical terms, risk levels relative to measures of histogram-valued data on VaR are calculated as an aid to decision-making. The empirical study was carried out on 1023 HFR-based hedge funds, where we had estimated monthly ARMA-GARCH or asymmetric GARCH VaR and CVaR measures between 01 January 2003 and 31 December 2008. We identify two sub-periods: from 2003 to 2005, and from 2006 to 2008 in order to identify a recovery period after the 2001–2002 crisis and the impact of the 2007–2008 crisis. First, the symbolic approach allows us to construct the measures of histogram-valued data on VaR by optimizing the definition of categories. A symbolic principal component analysis shows that the indices coming from the VaR of the GARCH and asymmetrical GARCH are the most pertinent. Second, we apply a criterion of inter-class inertia and retain a partitioning of hedge funds into three classes by dynamic K-means cluster analysis. For each of our sub-periods and for each class, a risk level is defined on the basis of the categories of the most discriminating variable.

Asset Liquidity Adjusted VaR

Liquidity risk is divided into asset liquidity risk and funding liquidity risk. The first one, known as market/product-liquidity risk arises when a forced liquidation of assets creates unfavorable price movements. This risk varies across categories of assets and across time as a function of prevailing market conditions. It can be managed by setting limits on certain markets or products and by means of diversification. Liquidity risk can be factored into VaR measures by ensuring the horizon is at least greater than an orderly liquidation period.

Tail VaR Measures in a Multi-period Setting

This paper studies a coherent acceptability measure which is a negative coherent risk measure, in a multi-period model. When a coherent acceptability measure changes according to new information in the market, a time consistency plays an important role. The usual strong time consistency gives too severe a multi-period Tail Value at Risk (Tail VaR) from a practical viewpoint. We study a weak type of time consistency and propose new multi-period Tail VaR measures.

Risk and Return in Retail Forex

Retail Forex as an online CFD market is generally known as a high risk area for traders. This study is focused on existing risks and return possibilities in this market. Risks which may threat traders are surveyed through Liquidity, Credit, and Control, plus Market risks. Value at Risk and Moving Standardizations are applied due to market risk and risk of loss, while Momentum and Reversal effects on hourly, 4 hour and daily horizons during last 10 years are examined. Findings are indicating that Market Risk is the main risk in Retail Forex, while market risk is mostly depended to market volatilities. Also results are suggesting about the considerable performance of Reversal effect in hourly time frame of Last ten years, while studding Value At Risk measures have shown that risk of loss is related to the size of positions, not initial deposit. As a result, risk of loss of Retail FX traders can be bigger than initial deposit, since VaR measures in Retail FX can be equivalent with none levered standard contracts in foreign exchanges.

The impact of global financial crisis on the dependence structure of equity markets and on risk management

In this work, we use a time varying copula model to investigate the impact of the global financial crisis on dependence between US and each of six major stock markets and on risk management strategies. The model is implemented with an AR-GARCH-t for the marginal distribution and the extreme value copula for the joint distribution, which allow taking into account non-linear dependence, tails behaviour and their development over time. We investigate whether there are significant changes in the time-varying dependence structure of market and in VaR and ES measures especially during global financial crises period. Empirical results show that market dependences between US, European and Brazilian markets tend to increase considerably during crisis period and this increase started around the beginning of 2008. In the other hand, market volatility registered record levels around the end of 2008 due to the increase of the degree of uncertainty in this period. As a consequence, investors will allow more amounts to cover against negative evolution of portfolio value.

Optimal reinsurance with concave ceded loss functions under VaR and CTE risk measures

Most of the studies on optimal reinsurance are from the viewpoint of the insurer and the optimal ceded functions always turn out to be convex. However reinsurance contracts always involve a limit on the ceded loss function in practice, thus it may not be enough to confine the analysis to the class of convex functions only. In this paper, we study the problem of optimal reinsurance under VaR and CTE optimization criteria when the ceded loss functions are in the class of increasing concave functions. By using a simple geometric approach, we prove that under the VaR optimization criterion, the quota-share reinsurance with a policy limit is always optimal, while the full reinsurance with a policy limit is optimal under the CTE optimization criterion. Some illustrative examples are presented.

Analysis of the discounted sum of ascending ladder heights

Within the Sparre Andersen risk model, the ruin probability corresponds to the survival function of the maximal aggregate loss. It is well known that the maximum aggregate loss follows a compound geometric distribution, in which the summands consist of the ascending ladder heights. In the present paper, we propose to investigate the distribution of the discounted sum of ascending ladder heights over a finite or an infinite-time intervals. In particular, the moments of the discounted sum of ascending ladder heights over a finite and an infinite-time intervals are derived in both the classical compound Poisson risk model and the Sparre Andersen risk model with exponential claims. The application of a particular Gerber–Shiu functional is central to the derivation of these results, as is the mixed Erlang distributional assumption. Finally, we define VaR and TVaR risk measures in terms of the discounted sum of ascending ladder heights. We use a moment-matching method to approximate the distribution of the discounted sum of ascending ladder heights allowing the computation of the VaR and TVaR risk measures. We conclude this paper with a numerical example illustrating different topics discussed in the paper.

A detailed comparison of value at risk estimates

This work investigates the performance of different models of value at risk. We include several methods (parametric, historical simulation, Monte Carlo, and extreme value theory) and some models to compute the conditional variance. We analyze several international stock indexes and examine two types of periods: stable and volatile periods. To choose the best model, we employ a two-stage selection approach. The result indicates that the best model is a parametric model with conditional variance estimated by an asymmetric GARCH model under Student's t-distribution of returns. This paper shows that parametric models can obtain successful VaR measures if conditional variance is estimated properly.