represents an opportunity. Investors should take the following general approach to evaluating sources of risk in their portfolios: All sources of risk should be divided into two components, market risk and uncorrected risk. This division is conceptually simple-project the returns of each investment on the returns of the market and estimate the beta, the coefficient that estimates the multiple of the market return that is to be expected from that investment. The market risk of the investment is contributed by the estimated beta times the market return; the uncorrected risk of the investment is contributed by what is left-that is, by the volatility of the investment return minus the market return times the investment beta. The return associated with this residual component, called alpha, is the holy grail of active investment management. This division is interesting for a number of reasons. First, the market component of risk should be expected to earn a market-determined risk premium. As emphasized earlier, such a premium is available essentially for free in the market-that is, without an investment management fee. The cost of the market risk premium is not a fee, but rather its usage of a scarce resource, the investor's limited appetite for exposure to market risk. Uncorrelated risk is just the opposite. The uncorrelated risk does not create additional exposure to market risk. In most portfolios it therefore contributes very little to portfolio risk. Sources of uncorrelated risk, on the other hand, generally require an active management fee. The challenge highlighted by the equilibrium theory is whether an investment manager can actually create a positive alpha, that is, an expected return greater than the fee the manager charges for uncorrelated risk. The only way an investor can rationally determine whether the fees charged by a manager are reasonable, and whether the returns are adequate, is to separately identify the market risk and the uncorrelated risk components of the investment. We have opened this chapter with the question, when do uncorrelated assets add value to a portfolio? This question is interesting because it immediately highlights the fact that adding value to a portfolio is a function not of risk characteristics per se, but rather of the relationship between expected excess return and risk. In equilibrium, there is no special value to uncorrelated assets; in fact, they do not deserve an expected excess return. However, as we will show, the issues raised by considering uncorrelated assets are of more general interest. The circumstances that can make uncorrelated assets attractive, an expected excess return greater than the equilibrium value, can also apply to assets with positive correlations. Thus, this discussion leads naturally to a consideration of when, at the margin, adding any investment activity adds value to a portfolio. And finally, we will see that the same risk and return trade-offs apply not only at the margin, but also to the more general problem of how to optimally size all positions in a portfolio. Perhaps someday the world will be such that all investors will understand the distinction between market risk and uncorrelated risk, they will monitor the divisions of these components of risk in their investments, and their behavior will force prices to adjust so that there is no excess return, no alpha, left to be found in sources of uncorrelated risk. If that happens, investing will become less interesting and there will be fewer avenues through which to add value to portfolios. Our view is that such a world has not yet arrived, and our search for alpha continues. We do find it interesting, however, to think about how close we are to such a