rest in their empirical applications. Factor models have numerous applications. Investment management professionals use factor models to quantify a portfolio's return and risk characteristics. For example, factor models have been used in portfolio risk optimization, performance evaluation, performance attribution, and style analysis. In addition to the variety of applications, factor models have served as a basis to estimate: II Average, or unconditional returns-explaining differences in returns across a universe of stocks at a particular point in time. II Expected, or conditional returns-forecasting the expected value of stock returns using historical information. II Variances and covariances of returns-explaining the systematic variations and comovements among stock returns. In this chapter, our focus is on applications of equity factor models for measuring risk. The rest of this chapter is organized as follows: II We present a simple example of an equity factor model. This example sets the stage for a more formal introduction to factor models presented later. II We present the basics of factor returns and exposures. We provide two examples of different types of exposure calculations. 11 We provide a taxonomy of equity risk factor models. We organize factor models by observed and unobserved factor returns. II We take a detailed look at the linear cross-sectional factor model. We present local and global factor models. Typically, global factor models incorporate country and currency factors whereas local factor models do not. II We turn our attention to measuring and identifying sources of risk in a factor model. This section begins with definitions of various aspects of portfolios, then presents numerous formulas used in calculating contributions to risk, and concludes with an example from PACE, Goldman Sachs' proprietary risk and return attribution platform. II Finally, we summarize the risk estimation process and show the various steps required to estimate a linear factor model in practice. SIMPLE EQUITY FACTOR MODEL: AH EXAMPLE______________ What are factor models and what should managers know about them? We address these questions with an example that involves a particular application of a factor model. Specifically, we are interested in measuring the risk of a portfolio of stocks. The risk statistic that we calculate, whether it's standard deviation or some measure of Value at Risk (VaR), depends on the covariance matrix of stock returns. Hence, our focus is on estimating this covariance matrix. Suppose that our current portfolio consists of four stocks and that all time is measured in months. To calculate the portfolio's covariance matrix for the following month, from t to t + 1, we would do the following: