Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting

In this note, we comment on the relevance of elicitability for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk.

Tools for testing the Solvency Capital Requirement for life insurance

Longevity risk is one of the major risks that an insurance company or a pension fund has to deal with and it is expected that its importance will grow in the near future. In agreement with these considerations, in Solvency II regulation the Standard formula furnished for calculating the Solvency Capital Requirement explicitly considers this kind of risk. According to the new European rules in our paper we suggest a multiperiod approach to evaluate the SCR for longevity risk. We propose a backtesting framework for measuring the consistency of SCR calculations for life insurance policies.

Extreme Value Theory versus traditional GARCH approaches applied to financial data: a comparative evaluation

Although stock prices fluctuate, the variations are relatively small and are frequently assumed to be normal distributed on a large time scale. But sometimes these fluctuations can become determinant, especially when unforeseen large drops in asset prices are observed that could result in huge losses or even in market crashes. The evidence shows that these events happen far more often than would be expected under the generalized assumption of normal distributed financial returns. Thus it is crucial to properly model the distribution tails so as to be able to predict the frequency and magnitude of extreme stock price returns. In this paper we follow the approach suggested by McNeil and Frey (2000) and combine the GARCH-type models with the Extreme Value Theory (EVT) to estimate the tails of three financial index returns DJI,FTSE 100 and NIKKEI 225 representing three important financial areas in the world. Our results indicate that EVT-based conditional quantile estimates are much more accurate than those from conventional AR-GARCH models assuming normal or Student’s t-distribution innovations when doing out-of-sample estimation (within the insample estimation, this is so for the right tail of the distribution of returns).

Enhancing and implementing a pairs trading strategy

This paper designs a pairs trading model with the intent to identify existing profitable market opportunities to invest, i.e. traditionally strong correlated stocks that have diverged from its historical norm. It comprises a broad literature review on this strategy whose relevant findings (strategy improvements) are contemplated in the model. The authors combine the statistical results of the model with a backtesting analysis in order to provide guidance on the best investment opportunities.

Credit default swap (CDS) prediction model & trading strategy

This project focuses on the study of different explanatory models for the behavior of CDS security, such as Fixed-Effect Model, GLS Random-Effect Model, Pooled OLS and Quantile Regression Model. After determining the best fitness model, trading strategies with long and short positions in CDS have been developed. Due to some specifications of CDS, I conclude that the quantile regression is the most efficient model to estimate the data. The P&L and Sharpe Ratio of the strategy are analyzed using a backtesting analogy, where I conclude that, mainly for non-financial companies, the model allows traders to take advantage of and profit from arbitrages.

Modeling Dependence Structure and Forecasting Portfolio Value-at-Risk with Dynamic Copulas

We study the asymmetric and dynamic dependence between financial assets and demonstrate, from the perspective of risk management, the economic significance of dynamic copula models. First, we construct stock and currency portfolios sorted on different characteristics (ex ante beta, coskewness, cokurtosis and order flows), and find substantial evidence of dynamic evolution between the high beta (respectively, coskewness, cokurtosis and order flow) portfolios and the low beta (coskewness, cokurtosis and order flow) portfolios. Second, using three different dependence measures, we show the presence of asymmetric dependence between these characteristic-sorted portfolios. Third, we use a dynamic copula framework based on Creal et al. (2013) and Patton (2012) to forecast the portfolio Value-at-Risk of long-short (high minus low) equity and FX portfolios. We use several widely used univariate and multivariate VaR models for the purpose of comparison. Backtesting our methodology, we find that the asymmetric dynamic copula models provide more accurate forecasts, in general, and, in particular, perform much better during the recent financial crises, indicating the economic significance of incorporating dynamic and asymmetric dependence in risk management.

Study and implementation of some quantitative trading models

Quantitative or algorithmic trading is the automatization of investments decisions obeying a fixed or dynamic sets of rules to determine trading orders. It has increasingly made its way up to 70% of the trading volume of one of the biggest financial markets such as the New York Stock Exchange (NYSE). However, there is not a signi cant amount of academic literature devoted to it due to the private nature of investment banks and hedge funds. This projects aims to review the literature and discuss the models available in a subject that publications are scarce and infrequently. We review the basic and fundamental mathematical concepts needed for modeling financial markets such as: stochastic processes, stochastic integration and basic models for prices and spreads dynamics necessary for building quantitative strategies. We also contrast these models with real market data with minutely sampling frequency from the Dow Jones Industrial Average (DJIA). Quantitative strategies try to exploit two types of behavior: trend following or mean reversion. The former is grouped in the so-called technical models and the later in the so-called pairs trading. Technical models have been discarded by financial theoreticians but we show that they can be properly cast into a well defined scientific predictor if the signal generated by them pass the test of being a Markov time. That is, we can tell if the signal has occurred or not by examining the information up to the current time; or more technically, if the event is F_t-measurable. On the other hand the concept of pairs trading or market neutral strategy is fairly simple. However it can be cast in a variety of mathematical models ranging from a method based on a simple euclidean distance, in a co-integration framework or involving stochastic differential equations such as the well-known Ornstein-Uhlenbeck mean reversal ODE and its variations. A model for forecasting any economic or financial magnitude could be properly defined with scientific rigor but it could also lack of any economical value and be considered useless from a practical point of view. This is why this project could not be complete without a backtesting of the mentioned strategies. Conducting a useful and realistic backtesting is by no means a trivial exercise since the \laws" that govern financial markets are constantly evolving in time. This is the reason because we make emphasis in the calibration process of the strategies' parameters to adapt the given market conditions. We find out that the parameters from technical models are more volatile than their counterpart form market neutral strategies and calibration must be done in a high-frequency sampling manner to constantly track the currently market situation. As a whole, the goal of this project is to provide an overview of a quantitative approach to investment reviewing basic strategies and illustrating them by means of a back-testing with real financial market data. The sources of the data used in this project are Bloomberg for intraday time series and Yahoo! for daily prices. All numeric computations and graphics used and shown in this project were implemented in MATLAB^R scratch from scratch as a part of this thesis. No other mathematical or statistical software was used.

Market Index Biases and Minimum Risk Indices

Markets, in the real world, are not efficient zero-sum games where hypotheses of the CAPM are fulfilled. Then, it is easy to conclude the market portfolio is not located on Markowitz"s efficient frontier, and passive investments (and indexing) are not optimal but biased. In this paper, we define and analyze biases suffered by passive investors: the sample, construction, efficiency and active biases and tracking error are presented. We propose Minimum Risk Indices (MRI) as an alternative to deal with to market index biases, and to provide investors with portfolios closer to the efficient frontier, that is, more optimal investment possibilities. MRI (using a Parametric Value-at-Risk Minimization approach) are calculated for three stock markets achieving interesting results. Our indices are less risky and more profitable than current Market Indices in the Argentinean and Spanish markets, facing that way the Efficient Market Hypothesis. Two innovations must be outlined: an error dimension has been included in the backtesting and the Sharpe"s Ratio has been used to select the"best" MRI

Market Index Biases and Minimum Risk Indices

Markets, in the real world, are not efficient zero-sum games where hypotheses of the CAPM are fulfilled. Then, it is easy to conclude the market portfolio is not located on Markowitz"s efficient frontier, and passive investments (and indexing) are not optimal but biased. In this paper, we define and analyze biases suffered by passive investors: the sample, construction, efficiency and active biases and tracking error are presented. We propose Minimum Risk Indices (MRI) as an alternative to deal with to market index biases, and to provide investors with portfolios closer to the efficient frontier, that is, more optimal investment possibilities. MRI (using a Parametric Value-at-Risk Minimization approach) are calculated for three stock markets achieving interesting results. Our indices are less risky and more profitable than current Market Indices in the Argentinean and Spanish markets, facing that way the Efficient Market Hypothesis. Two innovations must be outlined: an error dimension has been included in the backtesting and the Sharpe"s Ratio has been used to select the"best" MRI

Is it possible to utilize VIX for a profitable trading strategy of S&P 500 Index?

The goal of this research was to make an overall sight to VIX® and how it can be used as a stock market indicator. Volatility index often referred as the fear index, measures how much it costs for investor to protect his/her S&P 500 position from fluctuations with options. Over the relatively short history of VIX it has been a successful timing coordinator and it has given incremental information about the market state adding its own psychological view of the amount of fear and greed. Correctly utilized VIX information gives a considerable advantage in timing market actions. In this paper we test how VIX works as a leading indicator of broad stock market index such as S&P 500 (SPX). The purpose of this paper is to find a working way to interpret VIX. The various tests are made on time series data ranging from the year 1990 to the year 2010. The 10-day simple moving average strategy gave significant profits from the whole time when VIX data is available. Strategy was able to utilize the increases of SPX in example portfolio value and was able to step aside when SPX was declining. At the times when portfolio was aside of S it was on safety fund like on treasury bills getting an annual yield of 3 percent. On the other side just a static number’s of VIX did not work as indicators in a profit making way.

Modelación estocástica y trading algorítmico del spread entre acciones mediante procesos de reversión a la media

Las estrategias de inversión pairs trading se basan en desviaciones del precio entre pares de acciones correlacionadas y han sido ampliamente implementadas por fondos de inversión tomando posiciones largas y cortas en las acciones seleccionadas cuando surgen divergencias y obteniendo utilidad cerrando la posición al converger. Se describe un modelo de reversión a la media para analizar la dinámica que sigue el diferencial del precio entre acciones ordinarias y preferenciales de una misma empresa en el mismo mercado. La media de convergencia en el largo plazo es obtenida con un filtro de media móvil, posteriormente, los parámetros del modelo de reversión a la media se estiman mediante un filtro de Kalman bajo una formulación de estado espacio sobre las series históricas. Se realiza un backtesting a la estrategia de pairs trading algorítmico sobre el modelo propuesto indicando potenciales utilidades en mercados financieros que se observan por fuera del equilibrio. Aplicaciones de los resultados podrían mostrar oportunidades para mejorar el rendimiento de portafolios, corregir errores de valoración y sobrellevar mejor periodos de bajos retornos.

Valor en riesgo del portafolio de TES de los bancos colombianos

En este trabajo se realiza la medición del riesgo de mercado para el portafolio de TES de un banco colombiano determinado, abordando el pronóstico de valor en riesgo (VaR) mediante diferentes modelos multivariados de volatilidad: EWMA, GARCH ortogonal, GARCH robusto, así como distintos modelos de VaR con distribución normal y distribución t-student, evaluando su eficiencia con las metodologías de backtesting propuestas por Candelon et al. (2011) con base en el método generalizado de momentos, junto con los test de independencia y de cobertura condicional planteados por Christoffersen y Pelletier (2004) y por Berkowitz, Christoffersen y Pelletier (2010). Los resultados obtenidos demuestran que la mejor especificación del VaR para la medición del riesgo de mercado del portafolio de TES de los bancos colombianos, es el construido a partir de volatilidades EWMA y basado en la distribución normal, ya que satisface las hipótesis de cobertura no condicional, independencia y cobertura condicional, al igual que los requerimientos estipulados en Basilea II y en la normativa vigente en Colombia.

Reajuste al modelo de aprobación y asignación de sobregiros para gestión de riesgos en el Banco de la Producción

La gestión de riesgo es aplicar estrategias para evitar los mismos o reducir los costos generados por la ocurrencia de éstos, dicha gestión requiere de un análisis para identificarlos y medirlos en su probabilidad e impacto, planificar y poner en práctica estrategias de control para mitigarlos manteniendo una constante retroalimentación. El presente trabajo, titulado “Reajuste al modelo de aprobación y asignación de sobregiros para gestión de riesgos en el Banco de la Producción.”, se basa en el riesgo de crédito que asume el prestador, derivado de la posibilidad de que el prestatario incumpla la obligación y que puede ser mitigado mediante el uso de la valoración crediticia del sujeto de crédito o en el estudio de los ingresos del sujeto y su comportamiento o historial crediticio. Banco de la Producción cuenta con un modelo de aprobación y determinación de montos máximos de sobregiros para los cuenta corrientistas el cual, le permite un análisis de cliente con el fin de determinar la probabilidad de incumplimiento para la aprobación del sobregiros y considerar otras variables que le permiten un cálculo del monto máximo de autorización del sobregiro; todo esto va en búsqueda de la mitigación de determinado riesgo de crédito. Considerando que dichos modelos deben ser revisados y actualizados continuamente para garantizar la mitigación del riesgo de crédito, se realizó esta investigación mediante un procedimiento estadístico denominado Backtesting, el cual “es utilizado para validar la calidad y precisión de un modelo, mediante la comparación de los resultados reales con las medidas de riesgo generadas por el modelo” y de esta manera poder determinar las desviaciones del mismo y proponer las respectivas mejoras. A través de la aplicación del Backtestig, se obtuvo un ratio de la idea que se prueba, la frecuencia de oportunidad bajo las condiciones establecidas, el promedio de posiciones ganadoras y perdedores de esta prueba.

Pairs trading : aplicação no mercado acionário brasileiro

Esta dissertação estuda a aplicação da estratégia Pairs Trading no mercado acionário brasileiro. Envolve basicamente a identificação de pares de ações que tenham movimentos de preço semelhantes e posteriormente a operação do diferencial entre seus preços. É possível observar no mercado a existência de um valor de equilíbrio de longo prazo para alguns pares de ações e quando o diferencial divergir de certa quantidade pré-definida opera-se o diferencial no intuito de que o mesmo retorne ao seu valor de equilíbrio de longo prazo, ou seja, espera-se que ocorra uma reversão à média do diferencial. A metodologia para a identificação desses pares de ações que descrevem movimentos semelhantes vem do conceito de cointegração. Essa metodologia é aplicada sobre as ações do índice Bovespa de 04-Jan-1993 a 30-Jun-2005. Inicialmente é feita uma pré-seleção dos pares de ações via correlação, ou medida de distância. Para a seleção final é feito o teste de cointegração, onde é utilizado o método de Dickey-Fuller aumentado (Augmented Dickey-Fuller test – ADF) para verificar a existência de raiz unitária da série de resíduo da combinação linear do logaritmo dos preços. Após a seleção, os pares são simulados historicamente (backtesting) para se analisar a performance dos retornos dos pares de ações, incluindo também os custos operacionais.

Pairs trading: uma aplicação ao mercado acionário brasileiro

Neste trabalho, verificamos viabilidade de aplicação da estratégia de pairs trading no mercado acionário brasileiro. Diferentemente de outros estudos do mesmo tema, construímos ativos sintéticos a partir de uma combinação linear de preços de ações. Conforme Burgeois e Minko (2005), utilizamos a metodologia de Johansen para a formação dos pares a serem testados. Após a identificação de pares cointegrados, para assegurar a estacionaridade do ativo sintético contruído a partir da relação linear de preços das ações, utilizamos os testes DF-GLS e KPSS e filtramos àqueles que apresentavam raiz unitária em sua série de tempo. A seguir, simulamos a estratégia (backtesting) com os pares selecionados e para encontrar os melhores parâmetros, testamos diferentes períodos de formação dos pares, de operação e de parâmetros de entrada, saída e stop-loss. A fim de realizarmos os testes de forma mais realista possível, incluímos os custos de corretagem, de emolumentos e de aluguel, além de adicionar um lag de um dia para a realização das operações