12%-10%. ;6%i i 4%- "cc a? CC o) 2% -*-*-*-*-*-*-*- >-t-. 0% S£ "-t-t-T-T-T-T-T-T-t -2%---4%-- 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 Funding Ratio -♦-Payout = 0% 2.50% 5% " 7.50% -s-10% FIGURE 10.9 Required Returns for Maintaining Funding Status lead to a decrease in the funding ratio. The larger the payout and the lower the funding ratio, the larger the required return. The results highlight the need for large equity allocations (or allocations to bond indexes that have a higher duration than the liability index) for underfunded plans. Overfunded plans, on the contrary, can tolerate negative returns and still maintain their funding status. Actually, for overfunded plans the higher the payout ratio, the larger the negative return they can tolerate. This is true because a given payout decreases assets by a smaller percentage than liabilities when the plan is overfunded. While maintaining current funding status is a plausible objective for over-funded plans, underfunded plans will need to try to improve their funding ratios unless they can count on a contribution from the plan sponsor. We next look at the returns required to reach fully funded status for underfunded plans. In the appendix, we show that given an initial value for the funding ratio F0, the expected funding ratio at any time t is given by E0Ft = 1+4**] 1-P 1- ?o+p- 1 + E[RX] 1-P E[Rx] + p (10.14) Note that the expected funding ratio depends only on the average return, not on its volatility. In order to calculate the return required to reach fully funded status over a given horizon, we set the left-hand side in the above expression equal to 1, fix the horizon t, and find the value for E[R ] that satisfies the equality.8 Figure 10.10 shows the results for t = 10. For the underfunded plans, the required return to reach fully funded status in 10 years is obviously larger than the return required to maintain current funding status. The difference between these two rates of return is larger the lower the payout policy. :Since there is no analytical solution we use a numerical algorithm.