know exactly its future liabilities, giving rise to the noise term discussed earlier. For such a fund there is no risk-free strategy in the sense that no asset allocation exists that will lock in a certain rate of return on the surplus. The least risky strategy for this fund is to purchase a portfolio of bonds that represents the best guess about future liabilities and to invest the remainder into the risk-free asset. This strategy will yield the lowest surplus volatility of all possible strategies, and this volatility will equal the noise volatility.1 Therefore it is natural to evaluate other investment strategies relative to this least volatile of all strategies in terms of their risk/reward trade-off. This is precisely what the RACS does. At this point in the discussion, we find it helpful to consider a concrete example in order to illustrate how a fund may want to think about asset allocation. The example will allow us to put to use the theoretical concepts we have developed thus far. We will now introduce the return and risk assumptions at the foundation of the example to follow. ILLUSTRATION OF STATIC MODEL For the purpose of the calculations to follow, we use equilibrium return assumptions derived from the Black-Litterman model with no views based on historically estimated volatilities and correlations. It is important to note, however, that the calculations to follow can be performed with any return assumptions desired. In fact, sometimes it is useful to see how sensitive the results are to the specific assumptions used. For the present analysis, we choose to model liabilities with respect to the Lehman Long Government and Credit Index and noise. As of June 30, 2002, the duration of this index was about 10.5. We hypothetically consider a pension fund with a duration of liabilities of 12. This dictates the choice of |i = 12Ao.5 = 1.14. For our basic scenario, we arbitrarily consider a noise return of zero and a volatility of 2 percent. This number is perhaps easier to interpret in terms of a confidence interval: If the noise is normally distributed, a noise volatility of 2 percent implies that in each given period the (excess) return on the liability index is within 4 percent of the (excess) return of the levered Lehman Long Government and Credit Index with a probability of 95 percent. For the liability index with a duration of 12, the 4 percent interval on returns translates into a 4%/12 = 33 bps interval on the yield on the liability index. Table 10.1 summarizes the risk/return assumptions used in the analysis to follow. All numbers reflect an annual horizon. For now, let us focus only on U.S. equity (represented by the S&P 500 index), the Lehman Long Government and Credit Index, and the liability index. First, because the duration of the liability index was assumed to be larger than that of the Lehman index, its excess return is also higher. This implies that even for a fund with a surplus, an all-bond allocation is a losing strategy in the long run, since the 2This is true as long as there is no asset that is negatively correlated with the noise. If such an asset exists, the fund can achieve a surplus volatility lower than the noise volatility.