We study the dynamics and cross-sectional properties of the variance risk premia embedded in options on stocks and indices, approximated by the synthetic variance swap returns. Several important stylized facts and contributions arise. First, variance risk premia for indices are systematically larger (more negative) than for individual securities. Second, there are systematic cross-sectional differences in the price of variance in individual stocks. Linking variance swaps to firm size/book-to-market, and stock turnover characteristics, an investor gains access to several lucrative long-short strategies with Sharpe Ratios around 2.85. Third, principal component analysis reveals at most one important factor driving both stock and variance swap returns, which corresponds to the traditional market factor. For the remainder of the dynamics, the stock and its variance processes are nearly linearly independent. Fourth, we find the leverage effect through analysis of the relationship between the variance risk premium and stock to variance correlation. The systematic (market factor) part of the leverage effect provides additional evidence of the existence of one factor common to both variance swaps and stocks, but the contribution of the market risk premium to the total variance premium is very small. These findings stress the importance of using variance-based instruments in the portfolio of an investor.

We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the risk-neutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premiums on five stock indexes and 35 individual stocks.

In 1952, Harry Markowitz published "Portfolio Selection," a paper which revolutionized modern investment theory and practice. The paper proposed that, in selecting investments, the investor should consider both expected return and variability of return on the portfolio as a whole. Portfolios that minimized variance for a given expected return were demonstrated to be the most efficient. Markowitz formulated the full solution of the general mean-variance efficient set problem in 1956 and presented it in the appendix to his 1959 book, Portfolio Selection. Though certain special cases of the general model have become widely known, both in academia and among managers of large institutional portfolios, the characteristics of the general solution were not presented in finance books for students at any level. And although the results of the general solution are used in a few advanced portfolio optimization programs, the solution to the general problem should not be seen merely as a computing procedure. It is a body of propositions and formulas concerning the shapes and properties of mean-variance efficient sets with implications for financial theory and practice beyond those of widely known cases. The purpose of the present book, originally published in 1987, is to present a comprehensive and accessible account of the general mean-variance portfolio analysis, and to illustrate its usefulness in the practice of portfolio management and the theory of capital markets. The portfolio selection program in Part IV of the 1987 edition has been updated and contains exercises and solutions.

This book is a compilation of recent articles written by leading academics and practitioners in the area of risk-based and factor investing (RBFI). The articles are intended to introduce readers to some of the latest, cutting edge research encountered by academics and professionals dealing with RBFI solutions. Together the authors detail both alternative non-return based portfolio construction techniques and investing style risk premia strategies. Each chapter deals with new methods of building strategic and tactical risk-based portfolios, constructing and combining systematic factor strategies and assessing the related rules-based investment performances. This book can assist portfolio managers, asset owners, consultants, academics and students who wish to further their understanding of the science and art of risk-based and factor investing. Contains up-to-date research from the areas of RBFI Features contributions from leading academics and practitioners in this field Features discussions of new methods of building strategic and tactical risk-based portfolios for practitioners, academics and students

Portfolio risk forecasting has been and continues to be an active research field for both academics and practitioners. Almost all institutional investment management firms use quantitative models for their portfolio forecasting, and researchers have explored models' econometric foundations, relative performance, and implications for capital market behavior and asset pricing equilibrium. Portfolio Risk Analysis provides an insightful and thorough overview of financial risk modeling, with an emphasis on practical applications, empirical reality, and historical perspective. Beginning with mean-variance analysis and the capital asset pricing model, the authors give a comprehensive and detailed account of factor models, which are the key to successful risk analysis in every economic climate. Topics range from the relative merits of fundamental, statistical, and macroeconomic models, to GARCH and other time series models, to the properties of the VIX volatility index. The book covers both mainstream and alternative asset classes, and includes in-depth treatments of model integration and evaluation. Credit and liquidity risk and the uncertainty of extreme events are examined in an intuitive and rigorous way. An extensive literature review accompanies each topic. The authors complement basic modeling techniques with references to applications, empirical studies, and advanced mathematical texts. This book is essential for financial practitioners, researchers, scholars, and students who want to understand the nature of financial markets or work toward improving them.