20% Very Underfunded -■ 40% 50% 60% Equity Allocation -Underfunded --is--Exactly Funded FIGURE 1 0.2 Surplus "Return5 70% 80% 90% 100% Overfunded Very Overfunded the expected change in surplus is linearly increasing in the equity allocation. It is easy to show that this is the case whenever the expected return on equity is larger than that on bonds, and it should be quite intuitive as well. The other interesting result shown here is the minimum equity allocation needed for a fund to prevent the surplus from shrinking. In the appendix this is shown to be Hs Px-1 U M!-p) (10.10) v-E-v-B where u,B and u,£ are the total expected return on bonds and equity, respectively. Given our present assumptions, for an underfunded plan (funding ratio 0.8) the minimum equity allocation is a little over 40 percent; at lower equity allocations the deficit will grow on average. Overfunded plans, on the contrary, will see their surplus grow even for a zero equity allocation. The important thing to remember in interpreting these results is, of course, that we have so far abstracted from any payouts. For an underfunded plan, the presence of payouts will further increase the required equity allocation to prevent the surplus from shrinking. We will come back to this point later in the dynamic analysis. Up to this point, we have shown that higher equity allocations lead to (usually) higher funding risk as well as higher expected changes in the surplus. This trade-off between risk and return can be illustrated by plotting both quantities in the same graph. The resulting picture very much resembles an efficient frontier and is shown in Figure 10.3. To facilitate the following interpretation, we have not normalized by the current asset value in this figure. We consider three funds with liabilities of S100 and assets of $80, $100, and $120, respectively. Along each line plotted, the solid markers represent equity allocations ranging from 0 percent at the left to 100 percent at the right. Let's look at the line labeled "underfunded," which corresponds to a funding ratio of 0.8, in a little more detail. Again, we see that a minimum equity allocation of over 40 percent is needed for