return as before: that it consists of a (possibly levered) position in a long bond index and uncorrected noise. Furthermore, we shall assume that returns are independently lognormally distributed through time with the means, volatilities, and correlations shown in the beginning of this chapter. Using the above expressions for the dynamics of assets and liabilities, it is easy to see that the surplus is not affected by the payout structure p. This should be intuitive, since a payout reduces assets and liabilities by the same amount. In a mul-tiperiod setup, however, the surplus is less useful a measure than in a single-period setup, since the absolute value of assets and liabilities can fluctuate widely. A $10 million surplus is a comfortable cushion for a plan with a $50 million liability, but will not evoke the same comfort if the value of liabilities grows to $100 million. For this reason we will focus on the funding ratio as the measure of interest in this section. The funding ratio, as will become apparent soon, does depend on the payout structure. Using our setup, we shall attempt to answer the following questions: II For an underfunded plan, what return on assets in excess of the return on liabilities is necessary to (1) retain the original funding ratio and (2) reach fully funded status over a given horizon? II For a given initial funding ratio, payout policy, and bond/equity split, how does the probability of being underfunded vary with the horizon? Required Returns Given a payout structure p, what return will keep the funding ratio constant on average? Letting Ft = AjLt, we can write \-p \-p Et[Ft+1] = FtEt 1 + R-Aj+l 1+Kl,,+, 1 P (10.12) and defining Rx t = (1 + RA f)/(l + R ) - 1 as the return on assets in excess of the return on liabilities, we find that EtK.]-El[Fl4l-P) + P (10.13, L J pt To keep the funding ratio constant on average we require Et[Ft+1]=Ff Using the last expression we can easily calculate the required average return as a function of the initial funding ratio for a given payout policy. Figure 10.9 shows the results. A plan that is 80 percent funded and pays out 7.5 percent of its liability value in a given year must achieve a 2 percent return on assets in excess of the return on liabilities in order to keep its funding ratio constant. A return lower than that will