Technical Sector Fundamental Statistical   Time Series Cross Section Principal Component I (Cross Section) Principal Component II (Cross Section) Cross Section and Time Series FIGURE 20.2 The Relationship between Factors, Data, and Model Estimation cess return (over the risk-free rate) on a particular security and the return on the market portfolio. We assume that the number of securities totals N. The mathematical expression for the excess return on the ?zth security can be described by the following one-factor model: r (t) - At) = a (t) + p (t)[f"(t) - rf(t)} + e (t) (20.8) where r {t) = Total return on the nth security at time t ¥{t) = Return on a risk-free security at time t a (t) = Stock return's alpha for the nth return (alpha also represents the expected return on a stock that has zero correlation to the market) Pn(t) = Market beta (beta measures the covariation between the market and the security return) f"{t) = Return on a market portfolio at time t ejt) = Mean-zero disturbance term at time t Equation (20.8) describes how the excess return of the nth security varies over time with the return on the market portfolio, its uncorrelated expected value (alpha), and an idiosyncratic term. The factor return in this model is rm{t) - rf(t) and it represents the systematic component of the nth stock's return. The idiosyncratic component of the ?zth stock's return is given by a (t) + e (t). In practice, in order to estimate the risk of an asset or portfolio using the market model we must estimate the market beta. This is done via time series regression. For example, we may collect, say, monthly stock and market returns over the past five years. We then regress10 60 excess stock returns on a constant and 60 market portfolio returns (over the risk-free rate). This yields an estimate of alpha and the market beta. Beta measures the sensitivity between the nth stock's excess return and the market portfolio return over this five-year period. In addition to the estimates of alpha and the market beta, practitioners want to know how much of the variation in excess returns is explained by the variation in market returns. The R-squared statistic provides such a measure. Specifically, the R-squared provides a 10Due to the statistical properties of the stock's return and the market return, estimation may involve more than ordinary least squares.