V = Ce-(T-() (10A.2)

where C = Projected benefit payment as of time t r = Discount rate as of time t

Over a short period of time, we can evaluate changes in the value of the benefit obligation due to changes in our projected benefit, changes in the discount rate, and the passage of time:

, av,_ av, av, dV = -dC +-dr + dt dC dr dt fl0A3) = V--(T-t)Vdr+rVdt

As a consequence, we have that

^L = ^-(T-t)dr+rdt (10A.4)

V C

Put another way, the incremental percentage change in the value of the liability is a sum of three terms. The first term, dC/C, is the percentage change in the projected benefit payout and therefore represents our uncertainty in the benefit cash flow. The second term, -(T - t)dr, reflects the uncertainty in the value due to uncertainty in discount rates (the term -f T - t) is the duration of the cash flow as of time t), whereas the final term, rdt, reflects change in value due to passage of time.

In the context of a pension plan, the first term could be interpreted as changes in the PBO due to changes in the actuarial cash flow projections resulting from, for example, different mortality assumptions, early terminations, lump sums, plan amendments, and acquisition/divestiture activity. The second term could be interpreted as the actuarial gain/loss due to a change in the discount rate, whereas the final term could be interpreted as the interest cost for the pension plan.

More generally, one could consider a pension plan with a steady rate of projected benefit payments CT, in which case the value of the liability as of time t would be given by

V = \cTe-rT<J-t]dT (10A.5) t

As before, we can evaluate the incremental changes in the value of the benefit obligation due to changes in projected cash flows, changes in the term structure of discount rates, and the passage of time:

dV = -Ctdt+\[dCT -(T-t)CT +rTCT]e~rT(T~t]dT (10A.6)

Again, each of the terms in equation (10A.6) has a natural interpretation in economic terms. The first term, -Ctdt, corresponds to benefits paid during the in-