Publication: Aplicaciones de la teoría de valores extremos a la gestión del riesgo
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2005-05
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2005-06-24
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Abstract
La intención de esta tesis es conocer más sobre la gestión del riesgo por medio de una
metodología muy diferente de las técnicas estadísticas normalmente utilizadas: varianza y
correlación. La alternativa utilizada es la teoría de valores extremos, que se presenta como
el medio natural para cuantificar el riesgo en enocometría financiera.
La tesis se concentra en el riesgo. Hay diferentes interpretaciones de este concepto que
dan lugar a diversas metodologías para cuantificar su magnitud e impacto en diferentes
características de la econometría financiera. En la introducción de la tesis se discute la distinción entre incertidumbre y riesgo desde diferentes puntos de vista, teoría de la decisión
y gestión del riesgo. Se sigue con una definición formal del riesgo motivada por teoría de
la decisión pero consistente con la metodología usada en la gestión del riesgo. El riesgo
se puede cuantificar por medio de técnicas estadísticas. Se caracteriza por las colas de la
distribución de los datos, en particular por la verosimilitud de cualquier suceso que conlleve
una característica negativa. En econometría financiera esta definición de riesgo se denomina
normalmente “downside risk” y se asocia con la cola izquierda de la distribución de los
rendimientos.
El objetivo del segundo capítulo es dar medidas adecuadas para cuantificar el riesgo
en series financieras. Para conseguir esto, se aplican herramientas derivadas de la teoría de
valores extremos.
Todas estas medidas del riesgo recientemente consideradas en la literatura basadas en
valores extremos se caracterizan en la práctica por métodos de selección ad-hoc de los valores
extremos (5 %, 1 %, etc.) La principal contribución en el segundo capítulo es proponer
una definición formal para estos valores. Los valores extremos de una muestra aleatoria
simple de tamaño n de una distribución F se definen como las observaciones que exceden
cierto umbral y siguen una distribución Generalizada de Pareto (GPD) donde el “tail
index” de F juega un papel principal. El umbral es el estadístico de orden que minimiza
un estadístico del tipo de Kolmogorov-Smirnov entre la distribución empírica de las correspondientes
observaciones mayores y la correspondiente GPD. Para formalizar la definición
usamos un bootstrap semiparamétrico para contrastar la correspondiente aproximación por
la distribución Generalizada de Pareto. Finalmente, usamos nuestra metodología para cuantificar
el riesgo estimando el tail index (es decir, el ratio de decaimiento de la cola negativa),
y el Valor en Riesgo (VaR) de algunos índices financieros de los principales mercados de
acciones.
Una vez que el riesgo se define y es formalmente cuantificado el siguiente objetivo de la
tesis es analizar los mecanismos de transmisión del riesgo en diferentes marcos. El capítulo
III se dedica a la transmisión del riesgo en series temporales. El riesgo se mide por la
ocurrencia de observaciones de gran magnitud y el canal de transmisión es la dependencia
temporal que se encuentra en los valores extremos y que pueden originar el agrupamiento
de estas observaciones. En este contexto existe un parámetro, el “extremal index” que gobierna
la dependencia temporal en las observaciones más altas, y tal que su recíproco mide
el nivel de agrupamiento (clustering) en los extremos. La contribución de la tesis en este
capítulo comienza por redefinir este parámetro. La definición provee un sencillo e inmediato
método de estimación para el extremal index con interesantes propiedades estadísticas
como son la consistencia y la distribución asintótica gasussiana. La existencia de clustering
en las observaciones más grandes es una consecuencia de la transmisión del riesgo derivado
de la ocurrencia de sucesos extremos. Una contribución muy importante en esta parte
es la posibilidad de contrastar la transmisión del riesgo en series financieras mediante el
contraste del clustering en los valores extremos. Esta teoría contrasta con teorías fundadas
en modelos para la volatilidad que modelizan la dependencia condicional en los segundos
momentos.
El siguiente capítulo trata sobre la transmisión del riesgo entre mercados financieros.
El interés en esta sección radica en distinguir interdependencia entre mercados, que surge
de los lazos normales entre diferentes economías, de los efectos de contagio, originados por
unas conexiones que se hacen más fuertes en periodos de crisis. Para hacer esto, las nociones
de interdependencia y contagio se revisan. La contribución en este punto se basa en
nuevas definiciones para estos conceptos basados en propiedades de las funciones cópula y
en monotonicidad en las colas, que se usarán para analizar el contagio direccional (causalidad
entre extremos). Esto es posible gracias a una innovadora función cópula que se deriva
de la teoría de valores extremos multivariante. Esta cópula nos permite modelizar diferentes
patrones de dependencia entre las variables de acuerdo al estado de los mercados, por
ejemplo en mercados a la baja o en mercados al alza. Este modelo es suficientemente flexible
para describir asimetrías entre las variables de tal manera que el contagio direccional
se puede contrastar. El modelo se aplica para medir el fenómeno de vuelo hacia la calidad
(flight to quality), es decir, flujos de capital que salen de los mercados de acciones hacia los
mercados de bonos cuando los primeros afrontan periodos de crisis.
Finalmente el capítulo V esboza las lineas de investigación futuras que implican diferentes
aspectos del análisis del riesgo.----------------
The intention of this dissertation is to provide some insight about risk management by using a methodology far from the standard statistical techniques: variance and correlation. The alternative is Extreme Value Theory, that is presented as the natural setup to quantify risk in financial econometrics. The thesis concentrates on risk. There are different interpretations of this concept that result in diverse methodologies to quantify its magnitude and impact on different characteristics of financial econometrics. In the introduction of the thesis the distinction between uncertainty and risk is discussed, regarding the point of view: decision theory or risk management. It follows with a formal definition of risk motivated by decision theory but consistent with the methodology used in risk management. Risk can be quantified by means of statistical techniques. Risk is characterized by the tails of the distribution of the data, in particular by the likelihood of any event entailing a negative feature. In financial econometrics this definition of risk is usually denominated downside risk and is associated with the left tail of the distribution of returns. The aim of the second chapter is to provide reliable measures to quantify the risk found in financial sequences. In order to achieve this, standard tools of extreme value theory are applied. All the risk measures recently considered in the literature based on extreme values are characterized in practice by ad-hoc selection methods for the extreme values (5%, 1%, etc.) The main contribution in the second chapter is to propose a formal definition for these values. The extreme values of any random sample of size n from a distribution function F are defined as the observations exceeding a threshold and following a type of generalized Pareto distribution (GPD) involving the tail index of F. The threshold is the order statistic that minimizes a Kolmogorov-Smirnov statistic between the empirical distribution of the corresponding largest observations and the corresponding GPD. To formalize the definition we use a semiparametric bootstrap to test the corresponding GPD approximation. Finally, we use our methodology to quantify risk by estimating the tail index (ratio of decay of the negative tail), and the value at risk (VaR) of some financial indexes of major stock markets. Once risk is defined and formally quantified the following aim in the thesis is analyzing its transmission mechanisms in different settings. Chapter 3 is devoted to the transmission of risk in time series. The risk is measured by the occurrence of significant large observations and the transmission channel is the serial dependence found in the extreme values that can originate clustering of data. In this context there exists a parameter, the extremal index, that governs the serial dependence in the largest observations, and such that its reciprocal measures the level of clustering in the extremes. The contribution of the thesis in this chapter starts by redefining this parameter. This definition provides a straightforward estimation method for the extremal index with appealing statistical properties; consistency, and asymptotic gaussian distribution. The existence of clustering in the largest observations is a byproduct of the transmission of risk derived from the occurrence of the largest observations. An outstanding contribution in this part is the possibility of testing the transmission of risk in financial sequences by testing the clustering in the extreme values. This theory contrasts with theories founded on volatility models that claim that serial dependence found in financial series is due to the conditional dependence on second moments. The next chapter involves the transmission of risk between financial markets. The interest lies in this section on distinguishing interdependence between markets, that surges from regular links between economies, from contagion effects, originated by increasing links between the markets in crises periods. In order to do this, the notions of interdependence and contagion are revisited. The contribution of the authors lies on new definitions for these concepts based on copula properties and tail monotonicity, that will be used to analyze directional contagion (causality between extremes). This is possible due to an innovative copula function that is derived from the multivariate extreme value theory. This copula allows us to model different patterns of dependence according to the state of the markets, e.g. bear or bull markets. This model is sufficiently flexible to describe asymmetries between variables in such a way that directional contagion can be tested. The model is applied to the flight to quality phenomenon, outflows of capital from the stocks markets to the bonds markets when the first ones are facing crisis periods. Finally Chapter 5 sketches future lines of research involving different aspects of the analysis of risk.
The intention of this dissertation is to provide some insight about risk management by using a methodology far from the standard statistical techniques: variance and correlation. The alternative is Extreme Value Theory, that is presented as the natural setup to quantify risk in financial econometrics. The thesis concentrates on risk. There are different interpretations of this concept that result in diverse methodologies to quantify its magnitude and impact on different characteristics of financial econometrics. In the introduction of the thesis the distinction between uncertainty and risk is discussed, regarding the point of view: decision theory or risk management. It follows with a formal definition of risk motivated by decision theory but consistent with the methodology used in risk management. Risk can be quantified by means of statistical techniques. Risk is characterized by the tails of the distribution of the data, in particular by the likelihood of any event entailing a negative feature. In financial econometrics this definition of risk is usually denominated downside risk and is associated with the left tail of the distribution of returns. The aim of the second chapter is to provide reliable measures to quantify the risk found in financial sequences. In order to achieve this, standard tools of extreme value theory are applied. All the risk measures recently considered in the literature based on extreme values are characterized in practice by ad-hoc selection methods for the extreme values (5%, 1%, etc.) The main contribution in the second chapter is to propose a formal definition for these values. The extreme values of any random sample of size n from a distribution function F are defined as the observations exceeding a threshold and following a type of generalized Pareto distribution (GPD) involving the tail index of F. The threshold is the order statistic that minimizes a Kolmogorov-Smirnov statistic between the empirical distribution of the corresponding largest observations and the corresponding GPD. To formalize the definition we use a semiparametric bootstrap to test the corresponding GPD approximation. Finally, we use our methodology to quantify risk by estimating the tail index (ratio of decay of the negative tail), and the value at risk (VaR) of some financial indexes of major stock markets. Once risk is defined and formally quantified the following aim in the thesis is analyzing its transmission mechanisms in different settings. Chapter 3 is devoted to the transmission of risk in time series. The risk is measured by the occurrence of significant large observations and the transmission channel is the serial dependence found in the extreme values that can originate clustering of data. In this context there exists a parameter, the extremal index, that governs the serial dependence in the largest observations, and such that its reciprocal measures the level of clustering in the extremes. The contribution of the thesis in this chapter starts by redefining this parameter. This definition provides a straightforward estimation method for the extremal index with appealing statistical properties; consistency, and asymptotic gaussian distribution. The existence of clustering in the largest observations is a byproduct of the transmission of risk derived from the occurrence of the largest observations. An outstanding contribution in this part is the possibility of testing the transmission of risk in financial sequences by testing the clustering in the extreme values. This theory contrasts with theories founded on volatility models that claim that serial dependence found in financial series is due to the conditional dependence on second moments. The next chapter involves the transmission of risk between financial markets. The interest lies in this section on distinguishing interdependence between markets, that surges from regular links between economies, from contagion effects, originated by increasing links between the markets in crises periods. In order to do this, the notions of interdependence and contagion are revisited. The contribution of the authors lies on new definitions for these concepts based on copula properties and tail monotonicity, that will be used to analyze directional contagion (causality between extremes). This is possible due to an innovative copula function that is derived from the multivariate extreme value theory. This copula allows us to model different patterns of dependence according to the state of the markets, e.g. bear or bull markets. This model is sufficiently flexible to describe asymmetries between variables in such a way that directional contagion can be tested. The model is applied to the flight to quality phenomenon, outflows of capital from the stocks markets to the bonds markets when the first ones are facing crisis periods. Finally Chapter 5 sketches future lines of research involving different aspects of the analysis of risk.
Description
Keywords
Riesgo financiero, Econometría