largely vanished after they were first discovered.18 3. Whether historical return premiums associated (statistically) with firm characteristics such as size and book-to-market ratios represent priced risk factors or are simply unexplained anomalies remains to be resolved. Daniel and Titman argue that the evidence suggests that past risk premiums on these firm- characteristic variables are not associated with movements in market factors and hence do not represent factor risk.19 Their findings, if verified, are disturbing because they provide evidence that characteristics that are not associated with systematic risk are priced, in direct contradiction to the prediction of both the CAPM and ICAPM. Indeed, if you turn back to the box in the previous chapter on page 276, you will see that much of the discussion of the validity of the CAPM turns on the interpretation of these results.     SUMMARY 1. A single-factor model of the economy classifies sources of uncertainty as systematic (macroeconomic) factors or firm-specific (microeconomic) factors. The index model as- sumes that the macro factor can be represented by a broad index of stock returns. 2. The single-index model drastically reduces the necessary inputs in the Markowitz port- folio selection procedure. It also aids in specialization of labor in security analysis. 3. According to the index model specification, the systematic risk of a portfolio or asset equals 2 2 and the covariance between two assets equals 2 . M i j M 4. The index model is estimated by applying regression analysis to excess rates of return. The slope of the regression curve is the beta of an asset, whereas the intercept is the as- sets alpha during the sample period. The regression line is also called the security char- acteristic line. The regression beta is equivalent to the CAPM beta, except that the regression uses actual returns and the CAPM is specified in terms of expected returns. The CAPM predicts that the average value of alphas measured by the index model re- gression will be zero. 5. Practitioners routinely estimate the index model using total rather than excess rates of return. This makes their estimate of alpha equal to rf (1 ). 6. Betas show a tendency to evolve toward 1 over time. Beta forecasting rules attempt to predict this drift. Moreover, other financial variables can be used to help forecast betas.       18 Fischer Black, "Beta and Return," Journal of Portfolio Management 20 (1993), pp. 8-18. 19 Kent Daniel and Sheridan Titman, "Evidence on the Characteristics of Cross Sectional Variation in Stock Returns," Journal of Finance 52 (1997), pp. 1-33. III. Equilibrium In Capital Markets 10. Single−Index and Multifactor Models The McGraw−Hill