On the Lp-Norm Regression Models for Estimating Value-at-Risk

Analysis of risk measures associated with price series data movements and its predictions are of strategic importance in the financial markets as well as to policy makers in particular for short- and longterm planning for setting up economic growth targets. For example, oilprice risk-management focuses primarily on when and how an organization can best prevent the costly exposure to price risk. Value-at-Risk (VaR) is the commonly practised instrument to measure risk and is evaluated by analysing the negative/positive tail of the probability distributions of the returns (profit or loss). In modelling applications, least-squares estimation (LSE)-based linear regression models are often employed for modeling and analyzing correlated data. These linear models are optimal and perform relatively well under conditions such as errors following normal or approximately normal distributions, being free of large size outliers and satisfying the Gauss-Markov assumptions. However, often in practical situations, the LSE-based linear regression models fail to provide optimal results, for instance, in non-Gaussian situations especially when the errors follow distributions with fat tails and error terms possess a finite variance. This is the situation in case of risk analysis which involves analyzing tail distributions. Thus, applications of the LSE-based regression models may be questioned for appropriateness and may have limited applicability. We have carried out the risk analysis of Iranian crude oil price data based on the Lp-norm regression models and have noted that the LSE-based models do not always perform the best. We discuss results from the L1, L2 and L∞-norm based linear regression models. ACM Computing Classification System (1998): B.1.2, F.1.3, F.2.3, G.3, J.2.

Modeling and forecasting value-at-risk for the Portuguese stock market

The aim of this work project is to find a model that is able to accurately forecast the daily Value-at-Risk for PSI-20 Index, independently of the market conditions, in order to expand empirical literature for the Portuguese stock market. Hence, two subsamples, representing more and less volatile periods, were modeled through unconditional and conditional volatility models (because it is what drives returns). All models were evaluated through Kupiec’s and Christoffersen’s tests, by comparing forecasts with actual results. Using an out-of-sample of 204 observations, it was found that a GARCH(1,1) is an accurate model for our purposes.

Nonparametric estimation of Value-at-Risk

A method to estimate an extreme quantile that requires no distributional assumptions is presented. The approach is based on transformed kernel estimation of the cumulative distribution function (cdf). The proposed method consists of a double transformation kernel estimation. We derive optimal bandwidth selection methods that have a direct expression for the smoothing parameter. The bandwidth can accommodate to the given quantile level. The procedure is useful for large data sets and improves quantile estimation compared to other methods in heavy tailed distributions. Implementation is straightforward and R programs are available.

Stress testing a complement of value-a-risk.

Rapport de stage (maîtrise en finance mathématique et computationnelle)

Estructura del portafolio de inversiones con el VAR (Valué at Risk)

El propósito general de este trabajo de investigación es el de identificar las herramientas que permitan evaluar riesgos, poder determinar un modelo de ayuda para la estructuración de portafolios y así retribuir al inversionista la mejor manera con un premio por riesgo en retorno de su inversión, Además de presentar un instrumento y demostrar las ventajas de su utilización en la valoración de riesgos en portafolios, se pretende distinguir los efectos económicos y financieros que el inversionista enfrenta. Para cumplir con este propósito, se realizo un diagnóstico y análisis de la actividad de los mercados Financieros y de Capitales, determinando los factores más importantes dentro de un modelo de valoración de riesgo para la estructura de un portafolio de renta variable, lo que me permitirá presentar de una manera clara, los aspectos técnicos y económicos que afectan a la estructura de una inversión aplicando la metodología denominada VAR (Valué at Risk); adicionalmente el manejo que se podría dar a las mismas para obtener un mayor beneficio. Los resultados obtenidos y su respectivo análisis constan a lo largo de este trabajo de investigación.