decreases. Let us see how modern portfolio theory addresses this question. In this example we will initially treat domestic and international equities as if they were the only two asset classes available for investment. In the absence of other constraints (transactions costs, etc.), optimal allocation of the risk budget requires equities to be allocated from domestic to international markets up to the point where the ratio of expected excess return13 to the marginal contribution to portfolio risk is the same for both assets. We focus on this marginal condition because it can provide guidance toward improving portfolios. Although a full-blown portfolio optimization is straightforward in this context, we deliberately avoid approaching the problem in this way because it tends to obscure the intuition and it does not conform to most investors' behavior. Portfolio decisions are almost always made at the margin. The investor is considering a purchase or a sale and wants to know how large to scale a particular transaction. The marginal condition for portfolio optimization provides useful guidance to the investor whenever such decisions are being made. This example is designed to provide intuition as to how this marginal condition provides assistance and why it is the condition that maximizes expected returns for a given level of risk. Notice that we assume that, at a point in time, the total risk of the portfolio must be limited. If this were not the case, then we could always increase expected return simply by increasing risk. Whatever the initial portfolio allocation, consider what happens if we shift a small amount of assets from domestic equities to international equities and adjust cash in order to hold the risk of the portfolio constant. In order to solve for the appropriate trades, we reallocate the amounts invested in domestic and international equities in proportion to their marginal contribution to portfolio risk. For example, if at the margin the contribution to portfolio risk of domestic equities is twice that of international equities, then in order to hold risk constant for each dollar of domestic equities sold we have to use a combination of proceeds plus cash to purchase two dollars' worth of international equities. In this context, if the ratio of expected excess returns on domestic equities to international equities is less than this 2 to 1 ratio of marginal risk contribution, then the expected return on the portfolio will increase with the additional allocation to international equities. As long as this is the case, we should continue to allocate to international equities in order to increase the expected return on the portfolio without increasing risk. Let us adopt some notation and look further into this example. Let A be the marginal contribution to the risk of the portfolio on the last unit invested in an asset. The value of A can be found by calculating the risk of the portfolio for a given asset allocation and then measuring what happens when we change that allocation. That is, suppose we have a risk measurement function, Riskfd, f), that we use to compute the risk of the portfolio with an amount of domestic equities, d, and an amount of international equities, f. We use the notation Risk(d, f) to emphasize that different measures of risk 13Throughout this book when we use the phrase "expected excess return," we mean the excess over the risk-free rate of interest.