as well to ensure being able to pay its liabilities in every period. The remainder of this chapter focuses on both a static fone-period) setup, as well as a dynamic setup. In our static analysis, we extend the notion of a risk/return trade-off in the form of a Sharpe ratio to accommodate the presence of a liability stream. In the dynamic analysis, we investigate the effect of payouts on overall funding characteristics of a pension plan. STATIC ANALYSIS In the absence of any liabilities, investors care about the characteristics of the distribution of the returns on their assets. In the presence of liabilities, investors care about returns on both assets and liabilities, and on how they are correlated. In order to develop a measure to compare asset allocations in the presence of liabilities, let's first define a few quantities. Let us denote by Af and Lf the value of assets and liabilities, respectively, at time t. The surplus is given by St=At-Lt (io.3) and the funding ratio is given by %=y- (10-4) Thinking of a pension plan as a company, the surplus measure is the equivalent of the market value of equity of a public company: It is the value that would be left for the shareholders if the company used all of its assets to pay off all of its liabilities. The important caveat in this comparison is that while owners of public companies are subject to limited liability and therefore the market value of their equity cannot be negative, the surplus of a pension plan can be negative. Of course, a deficit cannot be carried on forever, since otherwise the plan will become insolvent at some point in time. This will be mitigated either by a contribution from the sponsor to the plan or by asset returns that exceed the returns on the liabilities. In this section, we assume that pension plans care about the return on the surplus instead of the return on assets alone. This assumption nicely fits the analogy of a pension plan with a public company whose managers are entrusted with maximizing the value of shareholder equity. Talking about the percentage return on the surplus is slightly tricky, however, because the surplus can be zero, and hence any change in the surplus would lead to an infinite return. Therefore, instead of focusing on the percentage return, we consider the dollar change in the surplus as the primary concern of a pension fund. When a pension fund cares about the change in the surplus, what are some of the quantities it may be interested in? For one thing, the fund will be interested in the expected change in the surplus, and whether it is positive (surplus is expected to grow or deficit is expected to decline) or negative (surplus is expected to decline or deficit is expected to grow). The fund may also be interested in the uncertainty in the change in the surplus. Finally, a pension fund may be interested in the