Preface 1. The approximation problem and existence of best approximations 2. The uniqueness of best approximations 3. Approximation operators and some approximating functions 4. Polynomial interpolation 5. Divided differences 6. The uniform convergence of polynomial approximations 7. The theory of minimax approximation 8. The exchange algorithm 9. The convergence of the exchange algorithm 10. Rational approximation by the exchange algorithm 11. Least squares approximation 12. Properties of orthogonal polynomials 13. Approximation of periodic functions 14. The theory of best L1 approximation 15. An example of L1 approximation and the discrete case 16. The order of convergence of polynomial approximations 17. The uniform boundedness theorem 18. Interpolation by piecewise polynomials 19. B-splines 20. Convergence properties of spline approximations 21. Knot positions and the calculation of spline approximations 22. The Peano kernel theorem 23. Natural and perfect splines 24. Optimal interpolation Appendices Index.

Approximation theory and methods, by M. J. D. Powell. Pp 339. £25 (hardcover), £8·50 (paperback). 1981. ISBN 0-521-22472-1/29514-9 (Cambridge University Press)

We investigate the predictive performance of various classes of value-at-risk (VaR) models in several dimensions—unfiltered versus filtered VaR models, parametric versus nonparametric distributions, conventional versus extreme value distributions, and quantile regression versus inverting the conditional distribution function. By using the reality check test of White (2000), we compare the predictive power of alternative VaR models in terms of the empirical coverage probability and the predictive quantile loss for the stock markets of five Asian economies that suffered from the 1997–1998 financial crisis. The results based on these two criteria are largely compatible and indicate some empirical regularities of risk forecasts. The Riskmetrics model behaves reasonably well in tranquil periods, while some extreme value theory (EVT)-based models do better in the crisis period. Filtering often appears to be useful for some models, particularly for the EVT models, though it could be harmful for some other models. The CaViaR quantile regression models of Engle and Manganelli (2004) have shown some success in predicting the VaR risk measure for various periods, generally more stable than those that invert a distribution function. Overall, the forecasting performance of the VaR models considered varies over the three periods before, during and after the crisis. Copyright © 2006 John Wiley & Sons, Ltd.

/pdf/evaluating-predictive-performance-of-value-at-risk-models-in-3w90dq642u.pdf

Evaluating predictive performance of value‐at‐risk models in emerging markets: a reality check

https://epge.fgv.br/we/MFEE/GerenciamentodeRisco/2012?action=AttachFile&do=get&target=Artigo43.pdf

Using extreme value theory to measure value-at-risk for daily electricity spot prices

Extreme value theory is concerned with the study of the asymptotic distribution of extreme events, that is to say events which are rare in frequency and huge in magnitude with respect to the majority of observations. Statistical methods derived from it have been employed increasingly in finance, especially for risk measurement. This paper surveys some of those main applications, namely for testing different distributional assumptions for the data, for Value‐at‐Risk and Expected Shortfall calculations, for asset allocation under safety‐first type constraints, and for the study of contagion and dependence across markets under conditions of stress.

/pdf/extreme-value-theory-in-finance-a-survey-v1z0ix6pts.pdf

Extreme value theory in finance: a survey

Advanced electronic trading platforms and index exchange-traded funds (ETFs) have an impact on the minimum-variance hedging of stock indexes with futures. Minimum-variance hedging may provide better out-of-sample hedging performance than a naiuve futures hedge, but only in markets without active trading of ETFs or advanced electronic communications networks. There is no evidence now to suggest that complex econometric models that include, e.g., time-varying conditional covariances and error correction can improve on the simple ordinary least squares hedge ratio. In markets with actively traded index ETFs and well-established electronic trading, no significant efficiency gains are apparent from any minimum-variance hedge.

Effectiveness of minimum-variance hedging

Value-at-risk (VaR) measures are generated using extreme value theory by modelling the tails of the return distributions of six Asian financial markets during the recent volatile market conditions. The maxima and minima of these return series were found to be satisfactorily modelled within an extreme value framework and the value at risk measures generated within this structure were found to be different to those generated by variance–covariance and historical methods, particularly for markets characterised by high degrees of leptokurtosis such as Malaysia and Indonesia.

Value-at-risk: Applying the extreme value approach to Asian markets in the recent financial turmoil

Portfolio insurance strategies

Dynamic futures-hedging ratios are estimated across seven markets using generalized models of the variance/covariance structure The hedging performances of the resultant dynamic strategies are then compared with static and naive strategies, both in- and out-of-sample Bayesian-adjusted hedge ratios also are employed as error purgers The empirical results indicate that the generalized dynamic models are well specified and that their use in determining optimal hedge ratios can lead to improvements in hedging performance as measured by the volatilities of the returns on the optimally hedged position © 2003 Wiley Periodicals, Inc Jrl Fut Mark 23:241–260, 2003

Futures hedging using dynamic models of the variance/covariance structure

A portfolio insurance strategy is a dynamic hedging process that provides the investor with the potential to limit downside risk while allowing participation on the upside so as to maximize the terminal value of a portfolio over a given investment horizon. First, this paper introduces the basic concepts and payoffs of a portfolio insurance strategy. Second, it describes the theory of alternative portfolio insurance strategies. Third, it empirically compares the performances of various portfolio insurance strategies during different markets and time periods. Fourth, it summaries the recent market developments of portfolio insurance strategies, especially in terms of the variations of features in CPPI investments. Finally, it addresses the impacts of these strategies on financial market stability.

Portfolio Insurance Strategies: Review of Theory and Empirical Studies

Portfolio selection models using variance, Value-at-Risk (VaR) and expected shortfall measures of risk are analysed, assuming differing underlying return distributions. The expected shortfall approach provides advantages relative to the VaR approach in terms of lower portfolio downside risks. Furthermore, using the extreme value distribution provides more insights for the investor relative to the empirical distribution. Analysing portfolio selection using these differing risk measures around the recent sub-prime mortgage problem period provides topical insights into the asset allocation process for the investor.

https://link.springer.com/content/pdf/10.1057%2Fjam.2008.15.pdf

Portfolio selection in an expected shortfall framework during the recent ‘credit crunch’ period

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