**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

**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 1.05
^{2}+ 100 x 1.05 + 100 = 315.25 and the future value of series B is 150 x 1.05^{2}+ 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 1.05
^{2}+ 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

You need to log in first to add your comment. ###### BRENDAMBITHE

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

###### odette

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

###### TammTamm

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

###### lisalett

I don't understand the lack of sample questions.

###### Metalpro

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

###### vixignus

yeah these are good

###### VikramJ

we need questions!

###### TiredHand

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

###### 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.

###### 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.

###### sgossett86

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

###### Chl4072

Not understand at all

How to find it by using TI..?

###### saika

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

###### shanshan34

please upload a video. this is a little confusing

thanks