Example Choosing the Right Duration of the Bond Portfolio

The last topic in the static analysis is choosing the duration of the bond portfolio. Of course, the more closely the duration of the asset portfolio matches that of the liability index the better, since it leads to better immunization against changes in liability

6Here we only considered the cases of no and full diversification. It can be shown that slightly underfunded plans may benefit from a small level of equity diversification at high equity allocations. In other words, these plans may see a small increase in the RACS by investing part of their equity outside the home country.

value. It should therefore come as no surprise that all funds will lose from investing in a bond index with a different duration. The bond index we consider is the Lehman Aggregate, which had a duration of about 4.3 as of June 30, 2002.

Figure 10.8 shows an efficient frontier graph like the one in Figure 10.3. In the top panel, the funding ratio is 0.8, and we see that in order to achieve the same return as in the base case, the fund must accept higher surplus risk when it chooses to invest in the Lehman Aggregate rather than the Lehman Long Government and Credit, against which liabilities are modeled. In the bottom panel the funding ratio is 1.5, and the same conclusion holds, but the loss from moving to an index with a lower duration is smaller.

These results are easiest to understand if we choose a particular bond/equity split (one of the highlighted points along the lines) and consider what happens

Underfunded

Underfunded

Dollar Risk vs. Liabilities

—U.S. Equity/Lehman Long Gov't/Credit —*—U.S. Equity/Lehman Aggregate

Dollar Risk vs. Liabilities

—U.S. Equity/Lehman Long Gov't/Credit —*—U.S. Equity/Lehman Aggregate

Overfunded

Overfunded

—U.S. Equity/Lehman Long Gov't/Credit —*—U.S. Equity/Lehman Aggregate

FIGURE 10.8 Effect of Shortening Duration of Bond Portfolio

—U.S. Equity/Lehman Long Gov't/Credit —*—U.S. Equity/Lehman Aggregate

FIGURE 10.8 Effect of Shortening Duration of Bond Portfolio when we change the fixed income 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.

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