Black was the first to point out the universal hedging property, which arises when all investors have the same degree of risk aversion and when wealth in each country equals that country's market capitalization. Black's equilibrium is a simple special case of the more general equilibrium model.5 Before jumping into the math, it is perhaps best to clarify first what the international CAPM model addresses, and what it does not address. In most versions of the model, including Black's, the term "currency" refers to real rates of exchange between the consumption bundles of different groups of investors. Thus, the theory does not include inflation risk, a potential cause of changes in the exchange rates of the nominal currencies that we generally think about in the real world. Another set of complexities of the real world that the universal hedging equilibrium does not address is the distribution of ownership of wealth across different countries, and the heterogeneity of investors' risk tolerances across countries. The theory takes these characteristics as inputs, and as noted earlier, one of the simplifying assumptions of Black's equilibrium is that investors in each country have wealth equal to the market capitalization of the domestic assets of their country. Another simplifying assumption in Black's model, which we will see is easy to relax, is that investors in all countries have the same degree of risk tolerance. The standard international CAPM equilibrium models also assume that the usual efficient markets conditions hold; there are no barriers to trade; and there are no capital controls, information barriers, or other costs that make investors prefer domestic assets. Finally, as in the domestic CAPM, these models assume a single, infinitesimal time period. These one-period models do not address the intertemporal risks that arise in a dynamic economy. Other academics have, of course, extended the results described here by relaxing various of these assumptions. As in Chapter 4, we consider first the simplest version of the model, a world in which there are only two currencies and two assets and investors solve a mean-variance portfolio optimization problem. We then address the general model. Consider a two-country world in which there are two risky assets, domestic equity in each country. We will refer to the two countries as the United States and Japan and we will later work out an example with parameters reflective of them. Denote the exchange rate between the two countries-that is, the number of units of Japanese currency per unit of U.S. currency-by X. Without loss of generality, assume that at the beginning of the investment period X has the value 1. In other words, at the beginning of the period one unit of a Japanese consumption bundle trades for one unit of a U.S. consumption bundle. At the end of the period, X has an uncertain value that gives the rate of exchange between units of consumption in 4Roll and Solnik (1977) is referenced earlier. The additional references are as follows: Solnik, Bruno H., 1974, "An Equilibrium Model of the International Capital Market," Journal of Economic Theory 8, 500-524; Adler, M. and B. Dumas, 1983, "International Portfolio Choice and Corporation Finance: A Synthesis," Journal of Finance 38, 925-984; and Grauer, E, R. Litzenberger, and R. Stehle, 1976, "Sharing Rules and Equilibrium in an International Capital Market under Uncertainty," Journal of Financial Economics 3, 233-256. 5Some have argued that Black's is not a very interesting special case because we have no reason, for example, to believe that investors all have the same risk aversion. While this concern is legitimate, risk aversion is very difficult to estimate, and so one might also argue that in the absence of evidence to the contrary, Black's special case is a reasonable place to start.