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##### Learning Outcome Statements

5.f. demonstrate the use of a time line in modeling and solving time value of money problems.

### Subject 6. The cash flow additivity principle

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.

1. BRENDAMBITHE: YOUR study notes are simply the best although this is my first time I find them easy to understand
2. odette: I really agree with you. I couldnt read QM from the text sent to me. But i am really enjoying reading it now.
3. TammTamm: I understand this now but it would be nice to have a couple of basic questions.
4. lisalett: I don't understand the lack of sample questions.
5. Metalpro: I totally agree with BRENDAMBITHE. The notes will surely help me a lot.
6. vixignus: yeah these are good
7. VikramJ: we need questions!
8. TiredHand: Analyst Notes describes it SO MUCH BETTER THAN THE CFA textbook.
9. MNSaleem: I need help here, i am confused that through which formula The Future Value of Series A is determined, as t = 3 i expect calculation like 100 x 1.05(3) + 100 x 1.05(2) + 100.
OR also i think formula for Future value of regular annunity should be used.

So please some one make me clear. Thanks.
10. ybavly: @MNSaleem

The diagram will make it clear. We are looking for the value AT t=3 not at the end of the year t=3.
This is still an ordinary annuity. The FV here happens on t=3 which is the first day of year 3. So, we simply add the \$100 because there was no time to reinvest it.
11. sgossett86: Using the BAII+ cf key makes solving uneven cash flows very simple.