term is assumed to have volatility o and to be uncorrelated with the bond index, but it may be correlated with other returns. When the current cash flow projections reflect all available information (and therefore represent a best guess as to future benefit payouts), the expected change in the benefit obligation due to changes in projected payouts is zero. Since the noise term reflects uncertainty about future payouts, we can assume that the noise term has a zero mean as long as the current projected payouts are equal to their expected values. The appendix contains a numerical example of how to pin down the parameters |3 and o from the balance sheet of a pension fund. EVALUATING INVESTMENT DECISIONS IN THE PRESENCE OF LIABILITIES In the absence of liabilities, alternative investment structures are often compared on the basis of their Sharpe ratios. The Sharpe ratio measures how much return in excess of a risk-free rate an investment offers for each unit of volatility: SR; = ^~Rf (10.2) where )ii and oi are the mean and volatility, respectively, of investment structure i. In other words, the risk and return of investments are evaluated relative to cash. The objective of maximizing the portfolio Sharpe ratio in an asset-only framework is theoretically well-founded. As was shown in Chapter 4, in a one-period model an investor who maximizes his or her utility over end-of-period wealth will choose the portfolio with the highest Sharpe ratio if the investor's utility function is quadratic (irrespective of the distribution of returns) or if returns are multivariately normally distributed (irrespective of the investor's utility function). In the context of an asset-liability framework, there are two shortcomings to measuring the trade-off between risk and return using the Sharpe ratio. First, the Sharpe ratio considers only the risk and return of assets and ignores the presence of any liability stream. As we will see, some investment structures are better suited to hedge against changes in the value of liabilities than others. This ability to hedge should be taken into account when evaluating an investment, but it is ignored by the Sharpe ratio. A second shortcoming of the Sharpe ratio in the present context is that it is really only a theoretically well-founded concept in a one-period model. The solution of the maximum Sharpe ratio portfolio to the optimization problem with quadratic utility does not obtain when the investor derives utility from intermediate consumption as well as from final wealth, even when the period utility function is of the quadratic form. Assuming that a pension fund cares only about the distribution of assets (or the surplus) at one future point in time seems inappropriate for at least two reasons. First, it is unclear how to choose the future date given that pension funds generally expect to remain in business indefinitely. Second, a pension fund will care about