between securities. Firm-specific events would in- clude new inventions, deaths of key employees, and other factors that affect the fortune of the individual firm without affecting the broad economy in a measurable way. We can summarize the distinction between macroeconomic and firm-specific factors by writing the holding-period return on security i as   ri E(ri) mi ei (10.1) where E(ri) is the expected return on the security as of the beginning of the holding period, mi is the impact of unanticipated macro events on the securitys return during the period, and ei is the impact of unanticipated firm-specific events. Both mi and ei have zero expected values because each represents the impact of unanticipated events, which by definition must average out to zero. We can gain further insight by recognizing that different firms have different sensitivi- ties to macroeconomic events. Thus if we denote the unanticipated components of the macro factor by F, and denote the responsiveness of security i to macroevents by beta, i, then the macro component of security i is mi iF, and then equation 10.1 becomes3 ri E(ri) iF ei (10.2) Equation 10.2 is known as a single-factor model for stock returns. It is easy to imagine that a more realistic decomposition of security returns would require more than one factor in equation 10.2. We treat this issue later in the chapter. For now, let us examine the simple case with only one macro factor. Of course, a factor model is of little use without specifying a way to measure the factor that is posited to affect security returns. One reasonable approach is to assert that the rate of return on a broad index of securities such as the S&P 500 is a valid proxy for the com- mon macro factor. This approach leads to an equation similar to the factor model, which is called a single-index model because it uses the market index to proxy for the common or systematic factor. According to the index model, we can separate the actual or realized rate of return on a security into macro (systematic) and micro (firm-specific) components in a manner sim- ilar to that in equation 10.2. We write the rate of return on each security as a sum of three components:     Symbol   1. The stocks expected return if the market is neutral, that is, if the markets excess return, rM rf, is zero i 2. The component of return due to movements in the overall market; i is the securitys responsiveness to market movements i (rM rf) 3. The unexpected component due to unexpected events that are relevant only to this security (firm specific) ei     The holding period excess return on the stock can be stated as