Outline

  • Abstract
  • Jel Classification
  • Keywords
  • 1. Introduction
  • 2. Forecasting Value-at-Risk and Daily Capital Charges
  • 3. Models for Forecasting Var
  • 3.1. Garch
  • 3.2. Egarch
  • 3.3. Gjr
  • 4. Stochastic Dominance
  • 4.1. Definitions and Hypothesis Formulation
  • 4.2. Re-Centering Functions
  • 4.3. More on Weakly Dependent Data
  • 4.4. Linton, Maasoumi and Whang’s Subsampling Test
  • 5. Data and Implementation of Tests
  • 5.1. Data Description
  • 5.2. Block Bootstrapping and Subsampling
  • 5.3. Daily Capital Charges (dcc) and Evaluation Framework: Stochastic Dominance
  • 6. Results
  • 7. Conclusions
  • References
  • Further Reading

رئوس مطالب

  • چکیده
  • کلیدواژه ها
  • 1. مقدمه
  • 2. پیش بینی ارزش در معرض ریسک و مخارج سرمایه ای روزانه
  • 3. مدل های پیش بینی ارزش در معرض ریسک (VaR)
  • 3.1. GARCH
  • 3.2. EGARCH
  • 3.3. GJR
  • 4. تسلط تصادفی
  • 4.1. تعاریف و تدوین فرضیه
  • 4.2. توابع تمرکزگرایی مجدد
  • 4.3. بیشتر بر روی داده های وابسته ضعیف
  • 4.4. آزمون نمونه برداری فرعی لینتون، معصومی و وانگ
  • 5. داده ها و پیاده سازی آزمون ها
  • 5.1. توصیف داده ها
  • 5.2. خود راه اندازی بلوک و نمونه برداری فرعی
  • 5.3 مخارج سرمایه ای روزانه (DCC) و چارچوب ارزیابی: تسلط تصادفی
  • 6. نتایج
  • 7. نتیجه گیری

Abstract

The Basel III Accord requires that banks and other Authorized Deposit-taking Institutions (ADIs) communicate their daily risk forecasts to the appropriate monetary authorities at the beginning of each trading day, using one of a range of alternative risk models to forecast Value-at-Risk (VaR). The risk estimates from these models are used to determine the daily capital charges (DCC) and associated capital costs of ADIs, depending in part on the number of previous violations, whereby realized losses exceed the estimated VaR. In this paper we define risk management in terms of choosing sensibly from a variety of risk models and discuss the optimal selection of the risk models. Previous approaches to model selection for predicting VaR proposed combining alternative risk models and ranking such models on the basis of average DCC, or other quantiles of its distribution. These methods are based on the first moment, or specific quantiles of the DCC distribution, and supported by restrictive evaluation functions. In this paper, we consider robust uniform rankings of models over large classes of loss functions that may reflect different weights and concerns over different intervals of the distribution of losses and DCC. The uniform rankings are based on recently developed statistical tests of stochastic dominance (SD). The SD tests are illustrated using the prices and returns of VIX futures. The empirical findings show that the tests of SD can rank different pairs of models to a statistical degree of confidence, and that the alternative (recentered) SD tests are in general agreement.

Keywords: - - - - - - -

Conclusions

In the spectrum of financial assets, VIX futures prices are a relatively new product. As with any financial asset, VIX futures are subject to risk. In this paper we analyzed the performance of a variety of strategies for managing the risk, through forecasting VaR, of VIX futures under the Basel II Accord.

The alternative strategies for forecasting VaR of VIX futures, and for managing financial risk under the Basel II Accord, are several univariate conditional volatility models, specifically GARCH, EGARCH and GJR, with each based on either the Gaussian and Student t distributions. The main criterion for choosing among the alternative strategies was minimizing average daily capital charges. In the paper we used a methodology based on stochastic dominance that permits partial ordering of strategies by accommodating the entire distribution of DCC values. This methodology provides a search for uniformly higher ranked volatility models, based on large classes of evaluation functions and the entire DCC distribution.

The main empirical findings of the paper are as follows: 1. The Gaussian models are generally preferred to their Student-t counterparts.

2. SD relations between DCC values produced by Gaussian models are generally not uniformly ranked. An analysis of CDFs and ICDF seems to show, however, that EGARCH provides DCC distributions with greater uncertainty, so the other models would be preferred. A lack of uniform rankings by SD also indicates that there exist special utility/evaluation functions that may provide complete, albeit subjective, rankings.

3. Within the class of Student-t distributions, EGARCH SD both GARCH and GJR, implying that EGARCH would be uniformly preferred to GARCH and GJR by a financial risk manager.

4. In general, a stochastic dominance criterion can be used to rank different models of VIX futures and distributions, as illustrated in the previous empirical results. Even in cases of no FSD and SSD, the tests provide additional information about the entire distribution over specific ranges.

5. The graphs of the CDFs of each pair of models allow a comparison globally for the whole distribution, and also locally for a given range of DCC values and probabilities. This allows more specific comparisons than previously afforded based on the mean and other moments of the relevant distributions.

In this paper we have not found an optimal model in the sense that it outperforms the other models during the whole sample period. On the other hand, we have restricted attention to a set of widely used, though not necessarily exhaustive set of forecasting models and distributions. This paper takes into account the number of violations as defined by the Basel Accord through the computation of DCC. Subsequent analysis would take into account if each model not only provides lower DCCs, but also satisfies the requirement of keeping the number of violations away from the red zone. For the VIX futures returns, all our models remain in the green zone.

The results of the paper suggest that further work is needed to compare, not only univariate models, but also combinations of models, such as based on the mean or median. This framework presented above should also be applied to a portfolio of assets to determine the usefulness of the stochastic dominance approach. This paper performed pairwise comparisons for a variety of models. The extension to comparisons among multivariate models is a topic for future research, as is a detailed analysis of the useful information that is contained in the CDF and ICDF.

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