Time Value of Money Concepts

Cash Flow Diagram: The Basic Concept

A cash flow diagram allows you to graphically depict the timing of the cash flows as well as their nature as either inflows or outflows.

Such a diagram is very easy to construct. We start with a simple horizontal time line....

...and then add arrows to represent the inflows (arrows pointing from the line) or outflows (arrows pointing to the line) of cash ...

Although this concept seems simple enough, it is not without controversy. My use of arrows differs from that used by some financial textbooks. In these texts you'll often find that an "up" arrow represents money received and a "down" arrow money paid out.

However, I like to think of the time line as a "stream" and as time goes on, you either put money into the stream or you take money out of the stream. The money you put into the stream is an outflow represented by an arrow pointing to the stream: the money is flowing out from you into the stream. The money you take out of the stream is an inflow represented by an arrow pointing away from the stream: the money is flowing out from the stream in to you.

If you find this confusing then it may be comforting to know that in the overall scheme of things I don't think how you draw your arrows really matters. Just as long as you understand what they represent.

However, be aware that on this site the diagrams will always be drawn in accordance with my "stream" metaphor.

The "Stream" Metaphor: Check your Understanding

Which of the following diagrams does not represent an initial cash inflow followed by four cash outflows? Click the radio button next to your choice to confirm the answer.






Perspective: A Practical Example

There are always two sides to a financial transaction: a borrower and a lender; a buyer and a seller; an investor and an investment.

You should keep in mind the dual nature of financial transactions when drawing a cash flow diagram. From who's perspective will it be drawn?

The example of a mortgage can illustrate this issue. From the borrower's perspective the transaction consists of a large cash inflow followed by a series of smaller cash outflows. The situation is exactly reveresed for the lender:

Note that the calculated amounts (present or future value , payment amount, interest rate, number of periods) will be the same regardless of the perspective from which the cash flow diagram is drawn. It simply helps in understanding and describing the problem to be conscious of the perspective from which it is viewed.

Also note that the angled line at the right end of the time line above is used to represent a continuation of time (and in this case also of payments) that cannot be shown because of space limitations.

Illustrating Interest Payments

The rule regarding how interest payments are illustrated is fairly flexible: sometimes I include interest payments in a cash flow diagram and sometimes I don't. Normally I don't.

I typically include interest payments when they help contrast the difference between two alternatives. For example, when we are comparing an investment with monthly compounding to one with quarterly compounding:

However, in most cases including the interest payments in the drawing does more harm than good; they end up confusing a diagram whose primary purpose is to provide a simplified respresentation of a complex problem:

If you think the drawing above is messy, imagine how you would draw a diagram to depict daily compounding. Or worse, continuous compounding.

So the rule of thumb regarding the inclusion of interest payments in a cash flow diagram is this: in general interest payments are presumed and not shown. Only when illustration of the interest payments serves a specific purpose should they be included in the diagram.

Enhancing the Cash Flow Diagram

Now that we know how to draw cash flows, we can embellish our diagram to make it more useful.

In general we want to add labels to our diagram but only to the point that they are helpful. Keep in mind that the purpose of the diagram is to illustrate a complex financial transacation as concisely as possible. Adding too much information only confuses the issue.

Here are some ideas for labels to include in a cash flow diagram:

  • Include the dollar amout for each inflow or outflow where possible; for annuities with large periodic payments, you may want to indicate the payment amount only once.
  • Identify an unkown value with a question mark and its name (e.g., "? PV" for an unknown present value). On this site unknown values are shown in red.
  • Use an arrow line from one cashflow to another cashflow to show an accumulation,discounting, or amortization.
  • Include the interest rate on the accumulation or discount line and show the compounding frequency.
  • Label the time increment (e.g., years, months, etc.).
  • Add consise text labels to explain things not evident in the diagram.

Here are some examples:

Ex 1. What is the present value of $10 to be paid in 3 years at 5% componded monthly?

In order to be clear it always helps to add the appropriate units to the time line (e.g., years). You should also label the interest rate i with the compounding frequency.

Ex 2. From the borrower's perspoective, what is the interest rate charged on a loan of $100 that requires four annual payments of $27 using annual compounding.

This example is an amortization of a loan and you need to find the interest rate i. Specifying $0 as the future value is optional.

Ex 3. How many years will it take for $10 to grow to $20 at 8% componded monthly?

Here n is the unknown so you value time units in reference to their relative relationship with n.

Final Thought

Don't ever discount the value of a graphic. (No pun intended.)

Its purpose is to illustrate a complex situation as succinctly as possible. If you can illustrate a complex problem concisely and precisely, you are better able to focus on its solution.

I know some people who consider it childish to draw pictures. Yet when I was at Ernst & Young, we made a lot of money conveying complex tax and financial strategies to high-level executives, almost always with the aid of graphics. There is no shame in drawing a picture to help solve a problem.