in a one-period model can be extended to a setup that also includes liabilities by focusing on the surplus instead of on assets alone. When the quantity of interest is the surplus, the notion of a Sharpe ratio can be extended to that of a risk-adjusted change in surplus (RACS). Using the RACS as the measure of optimality, we find that underfunded plans, those for whom the value of liabilities exceeds the value of assets, gain most from higher equity allocations, whereas diversification of the equity portfolio is most beneficial to overfunded plans. In a dynamic model with payouts it can be shown once again that underfunded plans must take on large equity allocations in order to improve their funding status. APPENDIX Choice ol Parameters lor Liability Modeling______ In the preceding discussion, we have modeled the value of liabilities as a sum of two parts-a bond, which reflects the best guess about future obligations, and a noise term, which reflects the uncertainty of the future payments. The return on the bond as well as its correlation with other assets can be calculated by discounting projected obligations by the current term structure. Alternatively, a publicly traded bond index can be used as a proxy, where the index is levered to match the duration of the liability stream. Mathe matically, where R = Total return on the liability index at time t Rft = Risk-free rate of return RB t = Total return on a bond index 8f = Noise term The parameter p reflects the duration of the liability relative to the specified bond index. As such, it reflects uncertainty in the value of liabilities due to changes in interest rates. The noise term (with assumed mean return r|, and volatility o ) reflects uncertainty in future payouts and is assumed to be uncorrelated with the bond index, although it may be correlated with other returns. To illustrate a methodology for choosing parameters, we consider the case of modeling the projected benefit obligation (PBO) for a corporate defined benefit pension plan. For a pension plan, the PBO reflects the actuarial present value of benefits attributed to employees to date. As such, the PBO is an actuarial measure of the pension liability that is based on a number of assumptions, including mortality rates, future salary growth, early retirement, lump sum payouts, and an actuarial discount rate. The changes in the PBO are typically disclosed in the company's 10-K filing in the section "Pensions and Other Postretirement Benefits." As a simplified case, one can evaluate the situation where the pension plan has a single benefit payment in year T. The present value V of the projected benefit payment as of time t will be given by the following equation: