sloping. Since equities and bonds are not perfectly correlated, however, a small allocation to equity in an otherwise all-bond portfolio may actually decrease overall volatility. When the funding ratio is very large, liabilities matter little in determining the surplus risk. We see this diversification effect in Figure 10.1 in the line labeled "very overfunded," which decreases initially before increasing. In the appendix, we show that for a given funding ratio the surplus risk is minimized when a fraction of assets equal to f L ^ o|-posoE, (109) oE+o|-2poBoE is invested in equity and the remainder in bonds, where o£ is the volatility of equity, Og is the volatility of bonds, and p is their correlation. Note that this expression is independent of the noise volatility. This should be intuitive since in the present setup neither bonds nor equity can be used to diversify away the uncorrelated noise. Furthermore, this expression is increasing in the initial funding ratio.4 A fund with a deficit is better off investing in bonds, because they offer a better hedge against changes in the value of liabilities, leading to a lower surplus volatility. A fund with a surplus, however, may want to invest in bonds up to a point so as to duration match the liabilities, which offers the best possible hedge against changes in the liability value. Beyond that point, the fund may be better off (in terms of minimizing surplus volatility) by investing an incremental dollar in equities rather than bonds due to the diversification effect between equities and bonds mentioned earlier. To understand this effect, consider the case in which the fund can actually invest in its liability index. An overfunded plan will then minimize its surplus volatility by investing an amount equal to the value of liabilities into the liability index (thereby eliminating liabilities completely from the asset allocation problem) and investing the remaining surplus into the volatility-minimizing portfolio of equities and bonds. So why do the other lines in the above graph show the smallest risk for an equity allocation equal to zero? The answer is simply that the graph shows only the range of equity allocations from 0 percent to 100 percent, and the smallest risk for the other funding ratios actually occurs for negative equity allocations. Finally, let us look at the line labeled "very underfunded." The surplus risk for this plan is very large, as should be intuitive. Furthermore, compared to the other lines in the graph this line is flatter (i.e., it varies less with the equity allocation). When the value of the assets is very small compared to the liabilities, exactly how these assets are invested matters less from a risk perspective. Having inspected the surplus risk emanating from various equity allocations, let us now turn to analyzing the expected change in surplus for the various plans in our example. Figure 10.2 shows the expected change in surplus relative to initial asset value as a function of the equity allocation. Two interesting facts emerge from this picture. First, for a given funding ratio 4This is true as long as ob > pO"e, which holds for the values in the example. If this inequality is reversed, the expression will be decreasing in the funding ratio.