the marginal contribution to risk, and therefore the implied excess return on commodities, is zero. We can solve for the risk-minimizing position by setting Ac = 0, or equivalently, solving for c such that (c oc2 + d od Oc pdt) = 0 where pdc is the correlation between commodities and domestic equity. Holding fixed the two-thirds weight in domestic equity, this risk-minimizing position in commodities is 10 percent. Thus, an important intuition that helps make sense of implied views is as follows: Holding fixed the weights in all other assets, there is a risk-minimizing position for each asset. Weights greater than that risk-minimizing position reflect positive implied views; weights less than that risk-minimizing position reflect bearish views. In terms of implied views, there is nothing special about positions greater than or less than zero; the neutral point is the risk-minimizing position. In a single-asset portfolio the risk-minimizing position is, of course, zero. More generally, however, the risk-minimizing position is a function of the positions, volatilities, and correlations of all assets in the portfolio. Moreover, in multiple-asset portfolios, the risk-minimizing position for each asset can be a positive or a negative value. We can use the correlations among assets and the risk-minimizing position to identify opportunities to improve allocations in portfolios. In multiple-asset portfolios, the risk-minimizing position will only be at zero for assets that are uncorrected with the rest of the portfolio. Such uncorrected assets are likely to be very valuable. Any asset or investment activity that is uncorrected with the portfolio, but also has a positive expected excess return, should be added to the portfolio. In addition to commodities, such uncorrected activities might include the active risk relative to benchmark of traditional active asset managers, certain types of hedge funds, active currency overlays, and global tactical asset allocation mandates. More generally, in the case of assets or activities that do have correlations with the existing portfolio and therefore that have nonzero risk-minimizing positions, any position that lies between zero and the risk-minimizing position is likely to represent an opportunity for the investor. Such positions are counterintuitive in the same sense that the 5 percent commodity position was. The implied view is opposite to the sign of the position. Typically investors hold positive positions because they have positive views, and vice versa. Whenever this is the case and the actual position is less than the risk-minimizing position, it makes sense to increase the size of the position. This situation is an opportunity because increasing the size of the position will both increase expected return and decrease risk. In terms of asset allocation, the counterintuitive positions described here are not very common. Most positions in asset classes are long positions (very few investors hold short positions in asset classes), most asset returns correlate positively with portfolio returns (commodities are an exception), and most assets are expected to have positive excess returns. More generally, though, we will see that when portfolios of securities are constructed with risk measured relative to a benchmark, such counterintuitive positions arise quite often. In this chapter we have taken the simple idea of modern portfolio theory-that investors wish to maximize return for a given level of risk-and developed some very interesting, and not particularly obvious, insights into the sizing of positions. We have tried to develop these ideas in a way that is intuitive and which can be used to help make portfolio decisions at the margin. We avoid the usual approach