Quantitative Methods: Basic Concepts

Reading 6. The Time Value of Money

Learning Outcome Statements

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

CFA Curriculum, 2020, Volume 1

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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.

User Contributed Comments 14

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YOUR study notes are simply the best although this is my first time I find them easy to understand


I really agree with you. I couldnt read QM from the text sent to me. But i am really enjoying reading it now.


I understand this now but it would be nice to have a couple of basic questions.


I don't understand the lack of sample questions.


I totally agree with BRENDAMBITHE. The notes will surely help me a lot.


yeah these are good


we need questions!


Analyst Notes describes it SO MUCH BETTER THAN THE CFA textbook.


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.



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.


Using the BAII+ cf key makes solving uneven cash flows very simple.


Not understand at all

How to find it by using TI..?


Analyst Notes, you should create a tutorial for using cashflows. a small video for the calculator functions would help tremendously! Thanks.


please upload a video. this is a little confusing