1652   0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% Equity Allocation " Very Underfunded --Underfunded .... -If..... Exactly Funded -A-Overfunded Very Overfunded FIGURE 10.1 Surplus Risk asset portfolio will grow more slowly than the liability index on average.2 Equities appear more attractive from this perspective. A second observation relates to the correlations between the series. Note that the liability index is more highly correlated with the Lehman Long Government and Credit Index than with equities. Therefore, bonds appear to be a better hedge against changes in the value of liabilities than equities. We will now examine this trade-off arising from higher allocations to equity more closely. Example: Surplus Risk, Expected Change, and the RACS Let us begin by looking at the surplus risk. Figure 10.1 plots the surplus risk as a fraction of asset value: t[ i+1J (10.8) A Along the horizontal axis we plot different equity allocations, ranging from 0 percent to 100 percent, with the remainder of the assets invested in the Lehman Long Government and Credit Index. Each line in the graph represents a different initial funding ratio.3 In order to interpret this graph, let's again compare the present case to the one without liabilities. In this case, the surplus simply equals the assets, and the quantity plotted would be the volatility of asset returns. Since equities are more 2As long as there are no payouts, an average return on assets that is lower than that on liabilities will lead to a decrease in the funding ratio. In the presence of payouts an overfunded plan can accept a lower return on assets than on liabilities and still maintain or grow its surplus and/or funding ratio. We will show this later in the chapter when we incorporate payouts into our setup. 3The funding ratios are 0.5, 0.8, 1, 1.5, and 2, respectively.