Kernel Conditional Quantile Estimation for Stationary Processes with Application to Conditional Value-at-Risk

The paper considers kernel estimation of conditional quantilesfor both short-range and long-range-dependent processes. Undermild regularity conditions, we obtain Bahadur representationsand central limit theorems for kernel quantile estimates ofthose processes. Our theory is applicable to many price processesof assets in finance. In particular, we present an asymptotictheory for kernel estimates of the value-at-risk (VaR) of themarket value of an asset conditional on the historical informationor a state process. The results are assessed based on a smallsimulation and are applied to AT&T monthly returns.     

Measuring large comovements in financial markets

A general, copula-based framework for measuring the dependence among financial time series is presented. Particular emphasis is placed on multivariate conditional Spearman's rho (MCS), a new measure of multivariate conditional dependence that describes the association between large or extreme negative returns—so-called tail dependence. We demonstrate that MCS has a number of advantages over conventional measures of tail dependence, both in theory and in practical applications. In the analysis of univariate financial series, data are filtered to remove temporal dependence as a matter of routine. We show that standard filtering procedures may strongly influence the conclusions drawn concerning tail dependence. We give empirical applications to two large data sets of high-frequency asset returns. Our results have immediate implications for portfolio risk management, derivative pricing and portfolio selection. In this context we address portfolio tail diversification and tail hedging. Amongst other aspects, it is shown that the proposed modeling framework improves the estimation of portfolio risk measures such as the value at risk.     

Risk management under extreme events

This article presents two applications of extreme value theory (EVT) to financial markets: computation of value at risk (VaR) and cross-section dependence of extreme returns (i.e., tail dependence). We use a sample comprised of the United States, Europe, Asia, and Latin America. Our main findings are the following. First, on average, EVT gives the most accurate estimate of VaR. Second, tail dependence of paired returns decreases substantially when both heteroscedasticity and serial correlation are filtered out by a multivariate GARCH model. Both findings are in agreement with previous research in this area for other financial markets.     

Value at risk estimation by quantile regression and kernel estimator

Risk management has attracted a great deal of attention, and Value at Risk (VaR) has emerged as a particularly popular and important measure for detecting the market risk of financial assets. The quantile regression method can generate VaR estimates without distributional assumptions; however, empirical evidence has shown the approach to be ineffective at evaluating the real level of downside risk in out-of-sample examination. This paper proposes a process in VaR estimation with methods of quantile regression and kernel estimator which applies the nonparametric technique with extreme quantile forecasts to realize a tail distribution and locate the VaR estimates. Empirical application of worldwide stock indices with 29 years of data is conducted and confirms the proposed approach outperforms others and provides highly reliable estimates.     

Nonparametric Estimation of Expected Shortfall

The expected shortfall is an increasingly popular risk measurein financial risk management and it possesses the desired sub-additivityproperty, which is lacking for the value at risk (VaR). We considertwo nonparametric expected shortfall estimators for dependentfinancial losses. One is a sample average of excessive losseslarger than a VaR. The other is a kernel smoothed version ofthe first estimator (Scaillet, 2004 Mathematical Finance), hopingthat more accurate estimation can be achieved by smoothing.Our analysis reveals that the extra kernel smoothing does notproduce more accurate estimation of the shortfall. This is differentfrom the estimation of the VaR where smoothing has been shownto produce reduction in both the variance and the mean squareerror of estimation. Therefore, the simpler ES estimator basedon the sample average of excessive losses is attractive forthe shortfall estimation.     

Risk Measurement Performance of Alternative Distribution Functions

This paper evaluates the performance of three extreme value distributions, i.e., generalized Pareto distribution (GPD), generalized extreme value distribution (GEV), and Box‐Cox‐GEV, and four skewed fat‐tailed distributions, i.e., skewed generalized error distribution (SGED), skewed generalized t (SGT), exponential generalized beta of the second kind (EGB2), and inverse hyperbolic sign (IHS) in estimating conditional and unconditional value at risk (VaR) thresholds. The results provide strong evidence that the SGT, EGB2, and IHS distributions perform as well as the more specialized extreme value distributions in modeling the tail behavior of portfolio returns. All three distributions produce similar VaR thresholds and perform better than the SGED and the normal distribution in approximating the extreme tails of the return distribution. The conditional coverage and the out‐of‐sample performance tests show that the actual VaR thresholds are time varying to a degree not captured by unconditional VaR measures. In light of the fact that VaR type measures are employed in many different types of financial and insurance applications including the determination of capital requirements, capital reserves, the setting of insurance deductibles, the setting of reinsurance cedance levels, as well as the estimation of expected claims and expected losses, these results are important to financial managers, actuaries, and insurance practitioners.     

A Bayesian encompassing test using combined value-at-risk estimates

The Value at Risk (VaR) is a risk measure that is widely used by financial institutions in allocating risk. VaR forecast estimation involves the conditional evaluation of quantiles based on the currently available information. Recent advances in VaR evaluation incorporate conditional variance into the quantile estimation, yielding the Conditional Autoregressive VaR (CAViaR) models. However, the large number of alternative CAViaR models raises the issue of identifying the optimal quantile predictor. To resolve this uncertainty, we propose a Bayesian encompassing test that evaluates various CAViaR models predictions against a combined CAViaR model based on the encompassing principle. This test provides a basis for forecasting combined conditional VaR estimates when there are evidences against the encompassing principle. We illustrate this test using simulated and financial daily return data series. The results demonstrate that there are evidences for using combined conditional VaR estimates when forecasting quantile risk.     

Multiresolution approximation for volatility processes

Abstract

We present an application of wavelet techniques to non-stationary time series with the aim of detecting the dependence structure which is typically found to characterize intraday stock index financial returns. It is particularly important to identify what components truly belong to the underlying volatility process, compared with those features appearing instead as a result of the presence of disturbance processes. The latter may yield misleading inference results when standard financial time series models are adopted. There is no universal agreement on whether long memory really affects financial series, or instead whether it could be that non-stationarity, once detected and accounted for, may allow for more power in detecting the dependence structure and thus suggest more reliable models. Wavelets are still a novel tool in the domain of applications in finance; thus, one goal is to try to show their potential use for signal decomposition and approximation of time-frequency signals. This might suggest a better interpretation of multi-scaling and aggregation effects in high-frequency returns. We show, by using special dictionaries of functions and ad hoc algorithms, that a pre-processing procedure for stock index returns leads to a more accurate identification of dependent and non-stationary features, whose detection results are improved compared with those obtained by other traditional Fourier-based methods. This allows generalized autoregressive conditional heteroscedastic models to be more effective for statistical estimation purposes.     

Portfolio selection based on the mean–VaR efficient frontier

Value-at-Risk (VaR) has become one of the standard measures for assessing risk not only in the financial industry but also for asset allocations of individual investors. The traditional mean–variance framework for portfolio selection should, however, be revised when the investor's concern is the VaR instead of the standard deviation. This is especially true when asset returns are not normal. In this paper, we incorporate VaR in portfolio selection, and we propose a mean–VaR efficient frontier. Due to the two-objective optimization problem that is associated with the mean–VaR framework, an evolutionary multi-objective approach is required to construct the mean–VaR efficient frontier. Specifically, we consider the elitist non-dominated sorting Genetic Algorithm (NSGA-II). From our empirical analysis, we conclude that the risk-averse investor might inefficiently allocate his/her wealth if his/her decision is based on the mean–variance framework.     

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.     

Semiparametric estimation of a characteristic-based factor model of common stock returns

We introduce an alternative version of the Fama–French three-factor model of stock returns together with a new estimation methodology. We assume that the factor betas in the model are smooth nonlinear functions of observed security characteristics. We develop an estimation procedure that combines nonparametric kernel methods for constructing mimicking portfolios with parametric nonlinear regression to estimate factor returns and factor betas simultaneously. The methodology is applied to US common stocks and the empirical findings compared to those of Fama and French.     

Monotonicity in asset returns: New tests with applications to the term structure,the CAPM,and portfolio sorts

Many theories in finance imply monotonic patterns in expected returns and other financial variables. The liquidity preference hypothesis predicts higher expected returns for bonds with longer times to maturity; the Capital Asset Pricing Model (CAPM) implies higher expected returns for stocks with higher betas; and standard asset pricing models imply that the pricing kernel is declining in market returns. The full set of implications of monotonicity is generally not exploited in empirical work, however. This paper proposes new and simple ways to test for monotonicity in financial variables and compares the proposed tests with extant alternatives such as t-tests, Bonferroni bounds, and multivariate inequality tests through empirical applications and simulations.     

Value at risk,cross-sectional returns and the role of investor sentiment

In this paper, we find that the relationship between the value-at-risk (VaR) and expected returns is negative and this negative relationship between the VaR and expected returns can be explained by volatility in the U.S. market. However, for different levels of investor sentiment, this relationship changes. For a high sentiment period, VaR is negatively related with the expected return and cannot be explained by momentum, short-term reversal, volatility, and financial distress. In comparison, the relation between the VaR and expected returns during a low sentiment period is mixed.     

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.     

Covariance dependent kernels,a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.     

Time varying betas and the unconditional distribution of asset returns

This paper draws attention to the fact that under standard assumptions the time varying betas model cannot capture the dynamics in beta. Conversely, evidence of time variation in beta using this model is equivalent to non-normality in the unconditional distribution of asset returns. Using the multivariate normal as a model for the joint distribution of returns on market indices and predetermined information variables, it is shown how to capture skewness and kurtosis in the unconditional distributions of asset returns. Under the assumptions of the model, asset returns are unconditionally distributed as an extended quadratic form (EQF) in normal variables. Expressions are given for the moment generating function and for the computation of the distribution and density functions. The market-timing model is derived formally using this model. The properties of bias when the standard linear betas model is used to estimate alpha when the correct model is the EQF are also investigated. It is shown that a different time varying betas model can arise as a consequence of portfolio selection. It is also shown that the predetermined information variables have the potential to account for the time series properties of returns, including heterogeneity of variance. An empirical study applies the model to returns on 46 UK bond funds. An analysis of the residuals shows that the model described in this paper is able to capture the dynamics of alpha and beta and properly account for other features of the time series of returns for 28 of these funds, of which 15 exhibit time variation in beta. The study reports the effect of the EQF model on the computation of VaR and CVaR and bias in the estimation of alpha.     

Model uncertainty and VaR aggregation

Despite well-known shortcomings as a risk measure, Value-at-Risk (VaR) is still the industry and regulatory standard for the calculation of risk capital in banking and insurance. This paper is concerned with the numerical estimation of the VaR for a portfolio position as a function of different dependence scenarios on the factors of the portfolio. Besides summarizing the most relevant analytical bounds, including a discussion of their sharpness, we introduce a numerical algorithm which allows for the computation of reliable (sharp) bounds for the VaR of high-dimensional portfolios with dimensions d possibly in the several hundreds. We show that additional positive dependence information will typically not improve the upper bound substantially. In contrast higher order marginal information on the model, when available, may lead to strongly improved bounds. Several examples of practical relevance show how explicit VaR bounds can be obtained. These bounds can be interpreted as a measure of model uncertainty induced by possible dependence scenarios.     

Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach

We propose a method for estimating Value at Risk (VaR) and related risk measures describing the tail of the conditional distribution of a heteroscedastic financial return series. Our approach combines pseudo-maximum-likelihood fitting of GARCH models to estimate the current volatility and extreme value theory (EVT) for estimating the tail of the innovation distribution of the GARCH model. We use our method to estimate conditional quantiles (VaR) and conditional expected shortfalls (the expected size of a return exceeding VaR), this being an alternative measure of tail risk with better theoretical properties than the quantile. Using backtesting of historical daily return series we show that our procedure gives better 1-day estimates than methods which ignore the heavy tails of the innovations or the stochastic nature of the volatility. With the help of our fitted models we adopt a Monte Carlo approach to estimating the conditional quantiles of returns over multiple-day horizons and find that this outperforms the simple square-root-of-time scaling method.     

El efecto de la volatilidad del peso mexicano en los rendimientos y riesgo de la Bolsa Mexicana de Valores

The effect of heavy tails due to rare events and different levels of asymmetry associated with high volatility clustering in the emerging financial markets requires sophisticated models for statistical modelling of such stylized facts. This article applies extreme value theory (EVT) to quantify tail risk on the daily returns of Mexican stock market under aggregation of foreign exchange rate risk from January 1971 to December 2010. This study focuses on the maximum-block method and generalized extreme value distribution (GEVD) to model the asymptotic behavior of extreme returns in US dollars. The empirical results show that EVT-Based VaR measured at high confidence levels performs better than simulation historical and delta-normal VaR models on capturing fat-tails in the returns of highly volatile stock markets. Additionally, international investors holding long positions in Mexican stock market are more prone to experience larger potential losses than investors with short positions during local currency depreciation and financial crisis periods.     

A Generalized Extreme Value Approach to Financial Risk Measurement

This paper develops an unconditional and conditional extreme value approach to calculating value at risk (VaR), and shows that the maximum likely loss of financial institutions can be more accurately estimated using the statistical theory of extremes. The new approach is based on the distribution of extreme returns instead of the distribution of all returns and provides good predictions of catastrophic market risks. Both the in-sample and out-of-sample performance results indicate that the Box–Cox generalized extreme value distribution introduced in the paper performs surprisingly well in capturing both the rate of occurrence and the extent of extreme events in financial markets. The new approach yields more precise VaR estimates than the normal and skewed t distributions.