of correlation coefficients can lead to nonsensical results. This can happen because some sets of correlation coefficients are mutually inconsistent, as the following example demonstrates:1     Correlation Matrix Standard Asset Deviation (%) A B C   A 20 1.00 0.90 0.90 B 20 0.90 1.00 0.00 C 20 0.90 0.00 1.00     Suppose that you construct a portfolio with weights: 1.00; 1.00; 1.00, for assets A; B; C, respectively, and calculate the portfolio variance. You will find that the portfolio variance appears to be negative ( 200). This of course is not possible because portfolio variances cannot be negative: we conclude that the inputs in the estimated correlation matrix must be mutually inconsistent. Of course, true correlation coefficients are always consistent.2 But we do not know these true correlations and can only estimate them with some imprecision. Unfortunately, it is difficult to determine whether a correlation matrix is inconsistent, pro- viding another motivation to seek a model that is easier to implement. Covariances between security returns tend to be positive because the same economic forces affect the fortunes of many firms. Some examples of common economic factors are business cycles, interest rates, technological changes, and cost of labor and raw materials. All these (interrelated) factors affect almost all firms. Thus unexpected changes in these variables cause, simultaneously, unexpected changes in the rates of return on the entire stock market.     1 We are grateful to Andrew Kaplin and Ravi Jagannathan, Kellogg Graduate School of Management, Northwestern University, for this example. 2 The mathematical term for a correlation matrix that cannot generate negative portfolio variance is "positive definite." III. Equilibrium In Capital Markets 10. Single−Index and Multifactor Models The McGraw−Hill Companies, 2001           294 PART III Equilibrium in Capital Markets     Suppose that we summarize all relevant economic factors by one macroeconomic indi- cator and assume that it moves the security market as a whole. We further