One of the most important considerations for a project manager is how the money flows in helping him/her determine whether the investment in a project is worthwhile or not. While there are many considerations beyond the strictly mathematical analysis of the time value of money, a quantitative model can be a first indication if an investment after a given period of time will turn a profit. As part of the process, it is necessary to convert the future value of money to a present value (likewise, one can determine what the future value of an investment, earning, or payment will be given its present value).

Time value of money can be calculated by the formulas listed in the table below:

Image from: https://en.wikipedia.org/wiki/Time_value_of_money.

Similar tables can be found in sources such as:* http://www2.fiu.edu/~keysj/TVM_(i).pdf).*

With the formulas given above, one can calculate the variable in the left column given the value in the second column, with the three unknown values being one of the following: Present Value, Future Value, and Fixed (or Repeating) Payment.

The different variables can be explained as follows:

P– Present value, that is, the value, time = 0 (such as for an investment or loan).

F– Future value, that is, the value at time = n.

A– Value of individual payments for each compounding period.

i– Interest rate. Note that this is an annual value. For compounding semi-annually, quarterly, monthly, weekly, or daily, divide this value by 2, 4, 12, 52, or 365, respectively. Other divisions are also possible.

n– Number of earning or payment periods. This value is the number of years if compounding is annual, but must be converted accordingly for other period types.

g– Growth rate for cash flow for each time period.

G– Growing payment amount that starts atGand increases byGeach period.

D– Exponentially or geometrically increasing payment amount that starts at an amountDand increases at a rate of (1 +g).

Some quick examples can illustrate the use of some of the formulas in the table. Suppose that you have a cash flow of $1000K (profit on a product) per quarter, and we consider this cash flow in Year 2, Quarter 2 of a product development. If the annual rate is 10 percent, the present value of this cash flow (Year 1, Quarter 1) can be found using Formula (2) from the table. We know:

F= $1000K

i= 10% per year, or 2.5% per quarter

n= 5 periods

Therefore, that $1000K cash flow will have a present value of $884K.

In another example, suppose you are negotiating the purchase of a wiggit and will be borrowing the money to pay for it. If the loan interest annually is 3.9 percent, and you want to pay back the loan over six years, but cannot afford a payment of more than $300/month, then (and assuming no down payments), the purchase price you need to negotiate can be found using Formula (6). We know:

A= $300

i= 3.9% per year, or 0.325% per month

n= 72 payment periods

** **Therefore, you will not be able to borrow more than $19231 (the present value of the loan).

Such formulas are useful in developing comprehensive economic models in product development. An overview of product development economics can be found references such as Ulrich and Eppinger [1]. In this textbook, Chapter 17 outlines how to set up a financial model for the product development process over a given period of time (divided into fiscal units such as quarters), and includes development costs, production ramp-up costs, production costs, and sales revenue, all of which are converted to present monetary values. Although assumptions are made to keep things simple, the model does help with the “go” or “no-go” decisions, and sensitivity analyses can be performed (i.e., answering “what if…” questions).

Having these formulas on hand will allow a project manager to make quick financial calculations under simplified conditions and gain some insight into the economic picture of a product development process.

### About the Author

**Dr. George Platanitis** has a Ph.D in Aerospace Engineering from Texas A&M University, an M.S. in Aerospace Engineering and Mechanics from University of Minnesota, and a B.S. in Engineering Physics from Queen’s University. His experiences include the areas of dynamics, control, aeroelasticity, and reliability. He is currently at Thales Canada Transport Solutions Inc. as a member of the RAM team. Outside of engineering, George’s interests include playing recreational hockey, and he has a passion for Salsa dancing.

###### Sources

*Karl T. Ulrich, Steven D. Eppinger, Product Design and Development, c. 2012 McGraw-Hill, New York.*