The Time Value of Money

At the heart of finance, accounting, economics, business, government, and families is MONEY. I do not mean it is the value system, the faith, or the reason for living. I’m talking strictly about the measurement system. Finance, accounting, and economics measure it or measure with it for their very existence as disciplines. Business, government, and families achieve whatever programs or adventures they wish to pursue generally with it. Business builds new plants or creates new product; government pursues the next best boondoggle; and families feed, clothe, shelter, educate, and entertain with it.

We focus on many financial matters in our classes including financial markets. Many of you may think the financial market sounds like a formidable place and that the concept must be terribly complicated. Actually it is amazingly simple. A financial market is a place where people come together to lend and borrow MONEY (people who need money meet up with people who have money). This is then the first characteristic or function of a financial market—it is a conduit between those participants who might want to lend (or invest) money at a particular time point and those who might want to borrow money at the same point. The lending and borrowing of money extend the financial market into the dimensions of time and uncertainty.

The basic second function of financial markets is setting the market interest rate, the price of money. Interest is a payment by the borrower to the lender for the use of the lender’s money during a period of time. Time is the first challenge to both parties: those who have funds and no preference to spend these funds NOW, and those without funds who find it appropriate, for whatever reason, to spend these funds NOW. The first party needs a place to protect their resources from inflation so that their money maintains value while they are not using it for consumption. The second party has a priority that makes the paying of some interest for the time they will use the resources worthwhile to them. The longer the borrower wants to use the lender’s funds the more compensation the lender will seek … generally.

Next the market (and lender) must consider the nature or the opportunity (the borrower) taking into consideration all sorts of variable to determine the risk (uncertainty) involved in the financial event. What is risk? It is the chances the lender will not recover his money?

When determining interest rates lenders consider two factors: (1) the risk free rate and (2) the risk premium. The formula for determining the interest rate is:

Interest (i) = Risk free rate (Rf) + Risk Premium (Rp)

Time can also play a role in determining the Rf and Rp. The most "risk-less" investments in our economy are U.S. government securities. The pricing of these securities contemplates time (you frequently hear quotes on 5-, 10-, and 30-year government bonds.) Time is what differentiates the interest rate on these investments as they are viewed as risk free. The Rp can also include additional "charges" for time risk.

So let’s move on to some math, the Time Value of Money.

Let’s assume that:

PV is a present value or the initial amount of loan

FV is a future amount (future value)

i equals the interest rate per time period

n is the number of time periods

If you borrow $200 for 1 year and agree to repay the lender $220 at the end of the year, the interest cost of borrowing is $20.

The formula to calculate the rate of interest (i) is:

$20/$200 = .10 or 10% per year.

All the mechanics of compound interest are illustrated in this simple example. There are two cash flows of importance here, the initial inflow of cash of $200 and the payment, 1 year later of $220. So using our variables, this example could be expressed as:

FV = PV + interest payment

And since the interest payment is usually expressed as a rate or percent (i) of the amount borrowed (PV) then:

FV = PV + (PV x i)

Which can be rearranged (simplified) in terms,

FV = PV (1 + i)

So (proof):

220 = 200 (1 + .10)

220 = 200 + 20

Now we can calculate the compounding over time by simply adding the "n" as an exponent to the equation.

FV = PV (1 + I)ⁿ

Now you can determine any one of these variables if you know the others. If you know the future value and the term (number of years or periods) and the interest rate you can determine the PV, initial or present value. It is the same with all the variables. Try it out with our example expanding the period to 3 years.

FV = $200 (1 + 10%)^3

First let’s do the exponent portion (1 + .10)³ that is 1.1 x 1.1 x 1.1 = 1.331, multiply that times $200 and you get $266.20. Why isn’t it $260? Because you get interest on the interest that is owed you. That is the nature of compounded interest versus simple interest. You can now insert any interest rate, term, and initial value to determine a future value.

You can calculate the present value (our initial value) of a future payment buy rearranging the same formula.

PV = FV / (1 + i)ⁿ .... FV divided by (1 + i)ⁿ

In our example:

PV = $266.20 / 1.331 = $200

All methods of time value analysis have their roots in this simple formula. Today calculators and computers do these calculations and the more complex events with monthly, quarterly, or yearly cash flows. This formula is at the heart of the computer formula and the other payment schedules.

One more easy adjustment. We have been calculating values "compounding annually," meaning the interest is calculated once per year. Many times today financial instruments compound quarterly, or monthly, or even daily. The adjustment to our formula is easy. You divide "i" by the number of periods in the year—for example, quarterly is by 4, monthly by 12, daily by 365—and you multiply "n" by the same factor.

So in our 3 year 10% example, if interest were compounded quarterly we would divide 10% by 4 = 2.5% and multiply 3 years by 4 = 12 periods. So instead of charging 10% interest for 3 years we would be charging 2.5% interest for 12 periods. So:

FV = $200 (1 + 2.5%)^12

FV = $200 x 1.34489

FV = $268.97

The added compounding generated $2.77 more interest over the 3 years. Not much, unless we were talking about $200 million and of course in the big picture we often are. Try doing the compounding monthly yourself and let's see what you get.

Here too, PV = FV/(1+i)^n ... try it PV = 268.97/1.34489 = $200, so the present value of $268.97 paid to you in 3 years if the discount rate was 10% compounded quarterly would be $200. On a Time Value of Money basis these two amounts $200 and $268.97 would be equal to one another if the $200 was paid to you today versus the $268.97 was to be paid to you in 3 years if the interest rate environment was 10% compounded quarterly.

Internal rate of return and net present value functions that determine the value of companies in acquisitions and the market, and the financial priority of vast numbers of capital projects in corporations all over the world are deeply rooted in this formula. They are all based on the time value of money.

Jerry Taylor is a full-time faculty member at Purdue Global. The views expressed in this article are solely those of the author and do not represent the view of Purdue Global.