the first term corresponds to a change in the benefit obligation due to adjustments in projected benefit payouts. The second integrand corresponds to a change in the benefit obligation due to changes in interest rates, whereas the final integrand corresponds to interest cost. Although the preceding model is simplified by looking only at incremental changes in value, it provides a connection to our methodology for modeling the noise term. In particular, the noise term is given by the expression dzf = _t__________________ Jcre-T^-^T (10A.7) When the current cash flow projections reflect all available information (and therefore represent a best guess as to future benefit payouts), we have that the expected change in the benefit obligation due to change in projected payouts is zero; that is, EJde] =0 (lOA.f Also, if we assume that the process &t has independent increments that are identically normally distributed, we have that Et[dzf\=v\dt where o is the instantaneous volatility of the noise process. (10A.9) MINIMIZING SURPLUS RISK FOR A GIVEN FUNDING RATIO Denoting the returns on equity and fixed income at time t as RE t and RB t, respectively, and the fraction of the surplus invested in equity as a, we write the surplus as St+1=At[a(l+REit+1) + (l-a)(l+RBit+1)]-Lt[l+Rf+^RBit+1-Rf) + £t+1] flOA.10) where we have used our model of the liability return. Dividing by the asset value Af we obtain L. L. L. A, = ccl+K£>f+1 +KBif+1 1-oe-^p -^ef+1+l-a-^[l+(l-P)Kf](10A.li; Our objective is to minimize the variance of this expression, or min Vart a V At J 2 2 = a Ge 1-a-p A ^2 t J r, ^2 KAtJ (10A.12)   2a 1-a-p A t J poEoB