would be one in which an asset is relatively independent of other investments in a portfolio and even though it may be risky by itself, it tends to add little to the overall risk of the portfolio. We refer to such investments as diversifiers, and we use them to increase return while living within an overall risk budget. Understanding and being able to measure and monitor the contribution to portfolio risk of every investment becomes a key part of the decision about how much to invest in each asset or investment activity. Assets that contribute less risk to a portfolio are less expensive in terms of using up the risk budget, and, everything else being equal, we should invest more in them. The intuition behind the mathematics that determines portfolio volatility can be seen in the geometry of a simple diagram. An asset affects the risk of a portfolio in the same way that the addition of a side to a line segment changes the distance of the end point to the origin. This nonlinear nature of adding risks, and the dependence on correlation, is illustrated in Figure 2.2. The length of the original line segment represents the risk of the original portfolio. We add a side to this segment; the length of the side represents the volatility of the new asset. The distance from the end of this new side to the origin represents the risk of the new portfolio. In the geometry of this illustration, it is clear how the angle between the new side and the original line segment is critical in determining how the distance to the origin is changed. In the case of portfolio risk, the correlation of the new asset with the original portfolio plays the same role as the angle between the new side and the original line segment. Correlations range between -1 and +1 and map into angles ranging from 0 to 180 degrees. The case of no correlation corresponds to a 90-degree angle. Positive correlations correspond to angles between 90 and 180 degrees, and negative correlations correspond to angles between 0 and 90 degrees. Let us consider a relatively simple example of how to use measures of contribution to portfolio risk to size investments and to increase expected returns. A key question that faces both individual and institutional investors is how much to invest in domestic versus international equities. One school of thought is that as c ............. £.**■. rr!"wiu!S!""";""" .... J _____________ ________ ^5___ Correlation Determines the Angle between A and B A = Old Portfolio Risk = New Investment Risk C= New Portfolio Risk FIGURE 2.2 Summation of Risk Depends on Correlation