326 research outputs found
Asymmetric multivariate normal mixture GARCH
An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out–of–sample Value–at–Risk measures
Multivariate Elliptical Truncated Moments
In this study, we derived analytic expressions for the elliptical truncated moment generating function (MGF), the zeroth, first, and second-order moments of quadratic forms of the multivariate normal, Student’s t, and generalised hyperbolic distributions. The resulting formulae were tested in a numerical application to calculate an analytic expression of the expected shortfall of quadratic portfolios with the benefit that moment based sensitivity measures can be derived from the analytic expression. The convergence rate of the analytic expression is fast – one iteration – for small closed integration domains, and slower for open integration domains when compared to the Monte Carlo integration method. The analytic formulae provide a theoretical framework for calculations in robust estimation, robust regression, outlier detection, design of experiments, and stochastic extensions of deterministic elliptical curves results
Asymmetric Multivariate Normal Mixture GARCH
An asymmetric multivariate generalization of the recently proposed class of normal mixture GARCH models is developed. Issues of parametrization and estimation are discussed. Conditions for covariance stationarity and the existence of the fourth moment are derived, and expressions for the dynamic correlation structure of the process are provided. In an application to stock market returns, it is shown that the disaggregation of the conditional (co)variance process generated by the model provides substantial intuition. Moreover, the model exhibits a strong performance in calculating out–of–sample Value–at–Risk measures.Conditional Volatility, Finite Normal Mixtures, Multivariate GARCH, Leverage Effect
Problem-driven scenario generation: an analytical approach for stochastic programs with tail risk measure
Scenario generation is the construction of a discrete random vector to
represent parameters of uncertain values in a stochastic program. Most
approaches to scenario generation are distribution-driven, that is, they
attempt to construct a random vector which captures well in a probabilistic
sense the uncertainty. On the other hand, a problem-driven approach may be able
to exploit the structure of a problem to provide a more concise representation
of the uncertainty.
In this paper we propose an analytic approach to problem-driven scenario
generation. This approach applies to stochastic programs where a tail risk
measure, such as conditional value-at-risk, is applied to a loss function.
Since tail risk measures only depend on the upper tail of a distribution,
standard methods of scenario generation, which typically spread their scenarios
evenly across the support of the random vector, struggle to adequately
represent tail risk. Our scenario generation approach works by targeting the
construction of scenarios in areas of the distribution corresponding to the
tails of the loss distributions. We provide conditions under which our approach
is consistent with sampling, and as proof-of-concept demonstrate how our
approach could be applied to two classes of problem, namely network design and
portfolio selection. Numerical tests on the portfolio selection problem
demonstrate that our approach yields better and more stable solutions compared
to standard Monte Carlo sampling
Regulatory Capital Modelling for Credit Risk
Abstract. The Basel II internal ratings-based (IRB) approach to capital adequacy for credit risk plays an important role in protecting the Australian banking sector against insolvency. We outline the mathematical foundations of regulatory capital modelling for credit risk, and extend the model specification of the IRB approach to a more general setting than the usual Gaussian case. It rests on the proposition that quantiles of the distribution of conditional expectation of portfolio percentage loss may be substituted for quantiles of the portfolio loss distribution. We present a more compact proof of this proposition under weaker assumptions. The IRB approach implements the so-called asymptotic single risk factor (ASRF) model, an asset value factor model of credit risk. The robustness of the model specification of the IRB approach to a relaxation in model assumptions is evaluated on a portfolio that is representative of the credit exposures of the Australian banking sector. We measure the rate of convergence, in terms of number of obligors, of empirical loss distributions to the asymptotic (infinitely fine-grained) portfolio loss distribution; and we evaluate the sensitivity of credit risk capital to dependence structure as modelled by asset correlations and elliptical copulas. A separate time series analysis takes measurements from the ASRF model of the prevailing state of Australia's economy and the level of capitalisation of its banking sector. These readings find general agreement with macroeconomic indicators, financial statistics and external credit ratings. However, given the range of economic conditions, from mild contraction to moderate expansion, experienced in Australia since the implementation of Basel II, we cannot attest to the validity of the model specification of the IRB approach for its intended purpose of solvency assessment. With the implementation of Basel II preceding the time when the effect of the financial crisis of 2007-09 was most acutely felt, our empirical findings offer a fundamental assessment of the impact of the crisis on the Australian banking sector. Access to internal bank data collected by the prudential regulator distinguishes our research from other empirical studies on the IRB approach and recent crisis
Facets of forecast evaluation
Forecasts are issued as point or probabilistic predictions, and their performance is measured using consistent scoring functions or proper scoring rules. We identify classes of elementary members of these performance measures and develop diagnostic tools to assist in the ranking of forecasters. Rankings are subject to the choice of evaluation criterion and the sampling variability. We also provide guidance in the computation of a particular scoring rule, the continuous ranked probability score
Problem-driven scenario generation:an analytical approach for stochastic programs with tail risk measure
Scenario generation is the construction of a discrete random vector to represent parameters of uncertain values in a stochastic program. Most approaches to scenario generation are distribution-driven, that is, they attempt to construct a random vector which captures well in a probabilistic sense the uncertainty. On the other hand, a problem-driven approach may be able to exploit the structure of a problem to provide a more concise representation of the uncertainty. There have been only a few problem-driven approaches proposed, and these have been heuristic in nature. In this paper we propose what is, as far as we are aware, the first analytic approach to problem-driven scenario generation. This approach applies to stochastic programs with a tail risk measure, such as conditional value-at-risk. Since tail risk measures only depend on the upper tail of a distribution, standard methods of scenario generation, which typically spread there scenarios evenly across the support of the solution, struggle to adequately represent tail risk well
Gini-type measures of risk and variability: Gini shortfall, capital allocations, and heavy-tailed risks
The final publication is available at Elsevier via http://dx.doi.org/10.1016/j.jbankfin.2017.06.013 © 2017. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/We introduce and explore Gini-type measures of risk and variability, and develop the corresponding economic capital allocation rules. The new measures are coherent, additive for co-monotonic risks, convenient computationally, and require only finiteness of the mean. To elucidate our theoretical considerations, we derive closed-form expressions for several parametric families of distributions that are of interest in insurance and finance, and further apply our findings to a risk portfolio of a bancassurance company.Natural Sciences and Engineering Research Council (NSERC) of Canada (Grant Numbers RGPIN-2016-356039, RGPIN-435844-2013, RGPIN-2016-427216)
EF and RZ also acknowledge the support of their research by the Casualty Actuarial Society (CAS
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