gross returns must be   1 rV rI IV (cV cI)     Our analysis so far has focused on the case of uncorrelated assets, allowing us to ignore issues of systematic risk. This special case turns out to be easy to generalize. If we were to allow for correlation among assets due to common systematic risk factors, we would find that the illiquidity premium is simply additive to the risk premium of the usual CAPM.19 Therefore, we can generalize the CAPM expected return beta relationship to include a liq- uidity effect as follows:   E(ri) rf i[E(rM) rf] f(ci) where f(ci) is a function of trading costs that measures the effect of the illiquidity premium given the trading costs of security i. We have seen that f(ci) is increasing in ci but at a de- creasing rate. The usual CAPM equation is modified because each investors optimal port- folio is now affected by liquidation cost as well as risk-return considerations. The model can be generalized in other ways as well. For example, even if investors do not know their investment horizon for certain, as long as investors do not perceive a con- nection between unexpected needs to liquidate investments and security returns, the impli- cations of the model are essentially unchanged, with expected horizons replacing actual horizons in equations 9.11 and 9.12. Amihud and Mendelson provided a considerable amount of empirical evidence that liq- uidity has a substantial impact on gross stock returns. We will defer our discussion of most of that evidence until Chapter 13. However, for a preview of the quantitative significance of the illiquidity effect, examine Figure 9.10, which is derived from their study. It shows that aver- age monthly returns over the 1961-1980 period rose from .35% for the group of stocks with the lowest bid-asked spread (the most liquid stocks) to 1.024% for the highest-spread stocks. This is an annualized differential of about 8%, nearly equal to the historical average risk pre- mium on the S&P 500 index! Moreover, as their model predicts, the effect of the spread on average monthly returns is nonlinear, with a curve that flattens out as spreads increase.     SUMMARY 1. The CAPM assumes that investors are single-period planners who agree on a common input list from security analysis and seek mean-variance optimal portfolios. 2. The CAPM assumes that security markets are ideal in the sense that: a. They are large, and investors are price-takers. b. There are no taxes or transaction costs. c. All risky assets are publicly traded. d. Investors can borrow and lend any amount at a fixed risk-free rate. 3. With these assumptions, all investors hold identical risky portfolios. The CAPM holds that in equilibrium the market portfolio is the unique mean-variance efficient tangency portfolio. Thus a passive strategy is efficient.