return to volatility-in sources of uncorrected risk. In equilibrium, of course, this ratio is zero. More generally, the larger this ratio, the more value uncorrected risk has in portfolio construction. Our view is that while markets are generally very efficient today, there is, nonetheless, still significant opportunity to create investment products with uncorrected risk having Sharpe ratios of .25 and higher, often much higher, after fees. As we will show, at such levels of the ratio of expected return per unit of risk the value of such products in portfolios is much greater than is generally understood, and the amount of uncorrelated risk that is optimal is much greater than that which is generally taken. Since, as we have highlighted, such an expectation for positive returns, much less returns greater than fees, is not an equilibrium phenomenon, perhaps we should explain why we think it exists. First, we believe most investors do not understand the distinction between market risk and uncorrelated risk. An important implication is that most investors have an aversion to uncorrelated risk that is not justified in equilibrium. This lack of understanding can create opportunities for investors willing to take advantage of them. A simple example of this phenomenon is provided by value stocks. Value stocks, those with low price-to-book and price-to-earnings ratios, tend to have lower than average betas, which in equilibrium would imply lower than average expected returns. Historically such stocks have actually provided higher than average returns. Second, we believe that not all information is public and fully digested by investors-the processing of information about relative values of assets is an expensive activity that requires resources, the allocation of capital, and exposure to risk. Those who initiate the purchases and sales that drive prices to fair value should be, and we believe are, compensated for their efforts. Third, there are noneconomic players in the marketplace, such as governments and central banks, which provide opportunities for profit-maximizing investors. Finally, there are many structural inefficiencies that prevent investors from driving risk premiums to their equilibrium values. These inefficiencies, such as higher than justified risk premiums in markets with barriers to foreign investors, again provide opportunities for those willing and able to take advantage of them. If these deviations from rationality and inefficiencies exist, then how does the CAPM help us to identify value? As we have seen in previous chapters, despite the fact that the "PM" in the acronym "CAPM" stands for "Pricing Model," in fact the CAPM does not price securities in the sense that it provides a level against which one can measure richness or cheapness. Rather, what the CAPM provides is a framework in which we can identify the equilibrium expected excess return for a security as a function of the risk characteristics of that security. In particular, the equilibrium expected excess return is a multiple of the beta of a security with the market portfolio. The equilibrium expected excess return should be interpreted as an economy-wide fair value for the degree of risk embedded in a security. It is not a function of the particular portfolio or situation of an individual investor. Even if the market does not cause all investments to yield an equilibrium risk premium, it is still useful to have such a neutral starting point from which an investor can then think about portfolio construction. If the equilibrium provides an "external" measure of value, one independent of the particular situation of the investor, then the investor's portfolio itself provides