benchmark from the Lehman Long Government and Credit Index to the Lehman Aggregate Index with a lower duration. First, since the expected return on the Lehman Aggregate is lower than that on the longer-duration index, the expected change in surplus will decrease, marked by a vertical downward shift in the graph. Second, since the Lehman Aggregate is a poor hedge for changes in liability value when compared with the Lehman Long Government and Credit, the surplus risk will increase. This is expressed by a horizontal shift to the right in Figure 10.8. The combined outcome of these two effects is, of course, a shift to the bottom right of each point along the line. For the overfunded plan, the vertical shift is higher than for the underfunded plan because there are simply more dollars changing benchmark, and the fund receives a lower expected return on each dollar. The horizontal shift, on the contrary, is larger the closer the fund is to fully funded status. When the fund is very underfunded, the hedging ability of the fixed income benchmark matters much less for surplus volatility than the absolute volatility of liabilities. When the fund is very overfunded, the presence of liabilities can almost be ignored, and what matters most is the absolute volatility of assets. In Figure 10.8, the fund on the top is closer to fully funded than the one on the bottom, and hence experiences the larger increase in volatility of the two. The above discussion implies that a fund is well served to invest in a bond index that is similar in duration to its liabilities. An additional issue that must be given consideration, however, is the difference in liquidity between short- and long-duration bonds. Large pension plans with long-duration liabilities will often find it impracticable to invest heavily in long-duration bonds, since the relatively low liquidity of these bonds impedes active trading. This issue is obviously more important the larger the pension fund, and it must be weighed with any return and hedging benefits from investing in long-duration bonds. DYNAMIC ANALYSIS Up to this point, we have investigated the asset allocation decision of a pension fund from a static point of view. We pretended that the fund had to make no payouts, and that it was concerned only with what happens to its surplus over one period, arbitrarily chosen to be one year. The setup was well suited to address many important issues like international diversification and the duration of the bond index to choose in the benchmark, but it leaves unanswered many important questions that affect pension funds in the long run. In this section we will look at a dynamic setup that will allow us to investigate the long-run impact of payouts. For analytical convenience we assume that the pension fund pays out a fixed fraction p of the value of liabilities at the end of each period.7 Mathematically, asset and liability values are assumed to evolve according to 7Some of the expressions we derive will not have closed-form solutions if we assume that the payout was made at the beginning of the period. We have performed various simulation exercises to gauge the quantitative impact of our assumption and have found that the numerical results are not at all sensitive to whether payouts are made at the beginning or end of the period. For this reason we have decided to stick with the more convenient setup.