The additivity principle: Dollar amounts indexed at the same point in time are additive.

Suppose we are considering two series of cash flows (A and B). The annual interest rate is 5%. We want to know the future value of combined cash flows at t = 3.

• We can calculate the future value of each series and add them up. The future value of series A is 100 x 1.05² + 100 x 1.05 + 100 = 315.25 and the future value of series B is 150 x 1.05² + 150 x 1.05 + 150 = 472.875. The future value of A + B is 788.125.

• Alternatively, we can add the cash flows of each series first and then find the future value of the combined cash flows: 250 x 1.05² + 250 x 1.05 + 250 = 315.25 = 788.125.

We can use this principle to solve many uneven cash flow problems if we add dollars indexed at the same point in time. Consider a cash flow series, A, with $100 indexed at t = 1, 2, 3 and 5, and $0 at t = 4. This series is an almost-even cash flow, flawed only by the missing $100 at t = 4. How do we find the present value of this series?

• We can create an annuity B with $100 indexed at t = 1, 2, 3, 4, 5. It's easy to find the present value of this series.
• Then we isolate an easily evaluated cash flow B - A; it has a single cash flow of $100 at t = 4. It's also easy to find the present value of this single cash flow.
• We then subtract the present value of B - A from the present value of B.