[v (£)] Explanation Inverse of market Weigh large-cap stocks more. Empirical research has shown capitalization that large-cap stocks have lower specific risk than small- cap stocks. Square root of inverse of Same as above, market capitalization Inverse of the volatility of Gives more weight to stocks that are better explained by the residual return from Capital Asset Pricing Model (CAPM). Residual is based market regression on regression from historical time period. By repeating the cross-sectional estimation each day over a period of time, say two years, we generate a time series of factor and specific returns. For example, suppose we run the cross-sectional regressions for T days (t = 1, . . . , T). Then we would have the T X K factor return matrix F?(T) where the fth row is a row vector of K elements representing the K factor returns at time t. In addition we would have a T X N specific return matrix, Uf(T), where the rth row is a vector of N-specific returns at time t. All risk calculations are based on covariance matrices of factor and specific returns. We obtain estimates of these covariance matrices using the data in F*{T) and U*(T), respectively. We describe the methods used to generate the covariance matrix estimates later in the chapter in the section on predicted factor and specific return covariance matrices. Factor-Mimic king Portfolios In this section we explain an interesting relationship between the regressions described earlier, and a particular trading strategy. Understanding this relationship facilitates the interpretation of factor returns. Factor returns generated from the cross-sectional regressions presented above are often described as returns to factor-mimicking portfolios. The term factor-mimicking portfolio comes from the idea that a portfolio of assets can be constructed in such a way that its behavior emulates the behavior of some factor. This portfolio is known as a long-short portfolio. A long-short portfolio consists of nearly equal amounts of long and short positions. Together, these positions have the ability to mimic particular factors. For example, a portfolio that consists of long positions in large-cap stocks and short positions in small-cap stocks is said to mimic the size factor. Large positive returns on such a portfolio show that large-cap stocks outperform small-cap stocks. Similarly, we can emulate the behavior of, say, the value factor by constructing a portfolio that is long assets with very high earnings-to-price (E/P) ratios (high value) and short assets with low E/P values (low value). High positive returns on such portfolios demonstrate that high-value stocks outperform low-value stocks. The reader may wonder how one can equate the return estimated from the cross-sectional regression specified by equation (20.34) and the return on a long-short portfolio. After all, they are both factor returns. Next, we show why the