....... <.... Exactly Funded Overfunded FIGURE 1 0.3 Surplus Risk and Return Trade-Off this plan to prevent the deficit from growing. But we also see that at this level of equity allocation, the risk versus the liabilities is about $7. In other words, a one standard deviation event would lead to an increase in the deficit of $ 7 to $27. The new funding ratio would be 0.73. Similarly, a two standard deviation event would lead to a funding ratio of 0.66. These numbers show the considerable risk underfunded plans face in attempting to reach fully funded status. The concept of risk-adjusted change in surplus (RACS) introduced earlier can be used to shed more light on how much a fund is earning in excess return for each unit of risk taken. Figure 10.4 graphs the RACS for the plans we have been discussing. Figure 10.4 clearly shows that for the underfunded and exactly funded plans the RACS is strictly increasing in the equity allocation, although for the underfunded plan the slope is steeper, implying that this plan is rewarded more for taking additional equity risk on a risk-adjusted basis. For the overfunded plan the story is quite different. The RACS increases very steeply early on but reaches its maximum at an equity allocation of around 30 percent. In order to understand this result, let's for a moment abstract from the presence of any noise. In this case, what strategy maximizes the RACS? A plan with sufficient funds can invest |3Lf in the bond index, perfectly hedging any future change in liabilities. Having thus basically eliminated liabilities from the asset allocation problem, the fund may use its remaining assets to purchase the portfolio that maximizes the Sharpe ratio of these assets.5 This case is illustrated in Figure 10.5, which plots the RACS for an overfunded plan (funding ratio 1.5) for different noise levels. What happens when we introduce noise? In this case, the ability of bonds to hedge changes in the liabilities is negatively impacted, and equity, with its higher Sharpe ratio, looks relatively more attractive. We therefore expect the optimal equity allocation (i.e., the allocation that maximizes the RACS) to increase. This can 5In the context of the numbers presented here, the portfolio of U.S. equity and the Lehman Long Government and Credit Index that maximizes the Sharpe ratio has an 83/17 bond/equity split.