Risk measures and their applications in asset management

Several approaches exist to model decision making under risk, where risk can be broadly defined as the effect of variability of random outcomes. One of the main approaches in the practice of decision making under risk uses mean-risk models; one such well-known is the classical Markowitz model, where variance is used as risk measure. Along this line, we consider a portfolio selection problem, where the asset returns have an elliptical distribution. We mainly focus on portfolio optimization models constructing portfolios with minimal risk, provided that a prescribed expected return level is attained. In particular, we model the risk by using Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). After reviewing the main properties of VaR and CVaR, we present short proofs to some of the well-known results. Finally, we describe a computationally efficient solution algorithm and present numerical results.conditional value-at-risk;elliptical distributions;mean-risk;portfolio optimization;value-at-risk

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing

Volatility forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly. JEL Klassifikation: C10, C53, G1

Copulas in finance and insurance

Copulas provide a potential useful modeling tool to represent the dependence structure among variables and to generate joint distributions by combining given marginal distributions. Simulations play a relevant role in finance and insurance. They are used to replicate efficient frontiers or extremal values, to price options, to estimate joint risks, and so on. Using copulas, it is easy to construct and simulate from multivariate distributions based on almost any choice of marginals and any type of dependence structure. In this paper we outline recent contributions of statistical modeling using copulas in finance and insurance. We review issues related to the notion of copulas, copula families, copula-based dynamic and static dependence structure, copulas and latent factor models and simulation of copulas. Finally, we outline hot topics in copulas with a special focus on model selection and goodness-of-fit testing.Dependence structure, Extremal values, Copula modeling, Copula review

Portfolio optimization when risk factors are conditionally varying and heavy tailed

Assumptions about the dynamic and distributional behavior of risk factors are crucial for the construction of optimal portfolios and for risk assessment. Although asset returns are generally characterized by conditionally varying volatilities and fat tails, the normal distribution with constant variance continues to be the standard framework in portfolio management. Here we propose a practical approach to portfolio selection. It takes both the conditionally varying volatility and the fat-tailedness of risk factors explicitly into account, while retaining analytical tractability and ease of implementation. An application to a portfolio of nine German DAX stocks illustrates that the model is strongly favored by the data and that it is practically implementable. Klassifizierung: C13, C32, G11, G14, G18Die Bewertung von Risiken und die optimale Zusammensetzung von Wertpapier-Portfolios hängt insbesondere von den für die Risikofaktoren gemachten Annahmen bezüglich der zugrunde liegenden Dynamik und den Verteilungseigenschaften ab. In der empirischen Finanzmarkt-Analyse ist weitestgehend akzeptiert, daß die Renditen von Finanzmarkt-Zeitreihen zeitvariierende Volatilität (HeteroskedastizitÄat) zeigen und daß die bedingte Verteilung der Renditen von der Normalverteilung abweichende Eigenschaften aufweisen. Insbesondere die Enden der Verteilung weisen eine gegenüber der Normalverteilung höhere Wahrscheinlichkeitsdichte auf ('fat-tails') und häufig ist die beobachtete Verteilung nicht symmetrisch. Trotzdem stellt die Normalverteilungs-Annahme mit konstanter Varianz weiterhin die Basis für den Mittelwert-Varianz Ansatz zur Portfolio-Optimierung dar. In der vorliegenden Studie schlagen wir einen praktikablen Ansatz zur Portfolio-Selektion mit einem Mittelwert-Skalen Ansatz vor, der sowohl die bedingte Heteroskedastizität der Renditen, als auch die von der Normalverteilung abweichenden Eigenschaften zu berücksichtigen in der Lage ist. Wir verwenden dazu eine dem GARCH Modellähnliche Dynamik der Risikofaktoren und verwenden stabile Verteilungen anstelle der Normalverteilung. Dabei gewährleistet das von uns vorgeschlagene Faktor-Modell sowohl gute analytische Eigenschaften und ist darüberhinaus auch einfach zu implementieren. Eine beispielhafte Anwendung des vorgeschlagenen Modells mit neun Aktien aus dem Deutschen Aktienindex veranschaulicht die bessere Anpassung des vorgeschlagenen Modells an die Daten und demonstriert die Anwendbarkeit zum Zwecke der Portfolio-Optimierung

Volatility Forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.

Volatility Forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3,4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.

Realized Betas and the Cross-Section of Expected Returns

What explains the cross section of expected returns for the 25 size/value Fama-French portfolios? It is found that modelling time-varying betas is important to explain the cross-section of expected returns, as well as to comply with the time series restriction on Jensen-alpha. Support for a modi?ed version of the conditional Jagannathan and Wang (1996) CAPM model is found, where implementation is carried out in the realized beta framework proposed in the paper. About 63% of the cross-sectional variability of the expected returns for the 25 Fama-French size and value sorted portfolios is then found to be explained by this parsimonious two-variable model.realized regression, time-varying beta, conditional CAPM

Volatility Forecasting

Volatility has been one of the most active and successful areas of research in time series econometrics and economic forecasting in recent decades. This chapter provides a selective survey of the most important theoretical developments and empirical insights to emerge from this burgeoning literature, with a distinct focus on forecasting applications. Volatility is inherently latent, and Section 1 begins with a brief intuitive account of various key volatility concepts. Section 2 then discusses a series of different economic situations in which volatility plays a crucial role, ranging from the use of volatility forecasts in portfolio allocation to density forecasting in risk management. Sections 3, 4 and 5 present a variety of alternative procedures for univariate volatility modeling and forecasting based on the GARCH, stochastic volatility and realized volatility paradigms, respectively. Section 6 extends the discussion to the multivariate problem of forecasting conditional covariances and correlations, and Section 7 discusses volatility forecast evaluation methods in both univariate and multivariate cases. Section 8 concludes briefly.