showing how much of portfolio volatility can be attributed, at the margin, to fixed income, equity, and foreign exchange (FX) positions. The graphs reveal clear patterns. First, when currency exposure is fully hedged, all portfolios have volatility of roughly 9 percent, with most of the risk attributable (at the margin) to the equity positions. Second, when none of the currency exposure is hedged, the fixed income positions contribute least to portfolio risk. Finally, currency positions are the greatest source of portfolio volatility for yen investors without currency hedging. Figure 11.5 suggests some flexibility across regions in setting currency hedging policies. For example, suppose that all investors want currency to contribute least to portfolio volatility (at the margin). According to Figure 11.5, U.S. dollar-based investors would hedge 40 percent of their currency exposure, while euro-based investors would hedge 80 percent. Now, how does home bias influence the currency hedging decision? Figure 11.6 plots currency's contribution to portfolio risk depending on the level of the currency hedge, the reference currency, and the degree of home bias (assuming that holdings are 32 percent fixed income and 68 percent equity). Each graph corresponds to a different reference currency, while one line reflects market capitalization weights and the other corresponds to a moderate, 60 percent diversified, "representative" degree of home bias. As a general rule, the greater the home bias, the lower the currency contribution to portfolio risk at each level of currency hedging. Clearly the risk associated with currency varies depending on the base currency and the degree of home bias. As a general rule, however, a 50 percent currency hedging policy will be sufficient to make currency a relatively small source of risk in the portfolio. Up to this point, we've focused on the risk associated with strategic currency positions. Viewed differently, we've specified allocations to domestic and international assets and levels of currency hedging, and then calculated portfolio volatility and the contribution to portfolio volatility of open foreign exchange positions. Little has been said about the returns associated with open currency positions. Because we're discussing currency in a portfolio context, our exploration of how hedging affects currency returns naturally focuses on the excess returns to currency-the returns an investor would receive above the returns embedded in the interest rate differentials (or currency forwards). We think implied returns analysis is a useful way to approach the issue of currency returns. Rather than assume explicit views on asset and currency returns to determine optimal portfolio weights, this method starts with a set of portfolio weights and determines what returns would optimize the portfolios.2 2Let x be an N x 1 vector of portfolio weights, Q be an N x N covariance matrix of asset returns (excess), and % a scalar risk aversion parameter. Then the N x 1 vector of returns R implied by the portfolio weights x is given by R = XClx. When x is the global capitalization weighted portfolio, then R is the vector of equilibrium returns. Notice that X can be calibrated so that portfolio excess returns are consistent with very-long-run historical experience.