Expected Excess Returns     Volatility Expecte d Excess Return Total Return Domestic equity International equity Cash 15% 16 0   5.5% 5.0 0.0 10.5% 10.0 5.0 investor starting with an equity allocation that is totally domestic. In order to generate a volatility of 10 percent the investor must hold a combination of cash plus domestic equity. In particular, given the assumed 15 percent volatility of domestic equity, the proportion allocated to equity is two-thirds of the total value and the allocation to cash is one-third of the total value. What happens as the investor starts to sell domestic equity and buy international equity? The marginal contributions to risk are simply the derivatives of this risk function with respect to the two arguments and can easily be shown to be given by the formulas: = d.o2d+f.od.of .p (2.9) d Riskfi, f) A _f-tf+d-od-QfP (2.10) f Risk(<i, f) In the special case when f = 0, these formulas simplify to: A,= d-a\n=Oj=.lS0 i ,, , \l/2 d2.o2 Af= (2 An =^.p = .104 d2.o2 Suppose the portfolio has a valuation, v, which is a large number, and an investor sells one unit of domestic equity; that is, let 5 = -1. Recalling equation (2.6), A (5)= Risk(i + 5, f)-Risk(i, f) 5 The risk of the portfolio is decreased by approximately: Risk(<i+5, /")-Risk(i, f) = .15 5 = -.15 (2.12) In order to keep risk unchanged, the investor must purchase A, .15 (2.13) = 1.442 Af .104