1.1 The Time Value of Money

There is a time value associated with money. That is, the value of a particular sum of money changes as time passes. Suppose that someone promised to give you a dollar. Would you rather have the dollar today or one year from today? Most likely, you would prefer to receive the dollar today for a variety of reasons.

First, by receiving the dollar today you ensure its receipt. Getting your money now removes the risk of not getting it in the future.

Second, if you have the dollar today, you have the option of using it today. You could invest it, pay your bills, or make a purchase. By delaying receipt of the dollar by a year, you lose the opportunity to use that dollar on any potential projects that may arise in the meantime.

Finally, inflation causes the value or purchasing power of a dollar to decrease. For example, an annual inflation rate of 2% means that an item that costs $1.00 today will cost about $1.02 in one year. Because inflation causes prices to rise over time, you can buy more with your dollar today than you can one year from now. The point is that all three of these forces (risk, opportunity, and inflation) reinforce one another to make a dollar today worth more than a dollar in the future. This relationship, known as the time value of money (TVM), is a key principle in financial decision-making.

Discount Rate

If we know that the passage of time affects the value of money, the next question is how much the value of money changes over a given period of time. To answer this question, imagine that you win a $1 million lottery but that you will not receive the money until one year from today (at which time you will receive the full amount). In order to avoid the three pitfalls of opportunity, risk, and inflation, what amount would you be willing to take today in exchange for the $1 million sometime in the future? For instance, would you accept $750,000 today in exchange for your future $1 million cash flow? If not, would you accept $850,000? Perhaps not, but the point is that there is an amount you would be willing to accept that is less than $1 million in exchange for $1 million in the future. The value that you would accept today is called the present value. Calculating present values (and, later, future values) is a key skill in finance. The difference between the present value and the future $1 million is determined by your personal attitudes and circumstances regarding risk, opportunity, and inflation. Each individual will have a different rate of return for which he or she will delay consumption through time and hence a different present value for the $1 million future payment.

When we think about the economy generally, we allow individual participants to “price” the rate at which they trade present consumption for future consumption by choosing among the set of financial securities—some have high risk and high expected return, while others have low risk and low expected return. Generally, the rate (or price) of moving consumption through time can be referred to as the discount rate (or interest rate) and can be written as follows:

$\text{Discount Rate}=\text{Risk-Free Rate}+\text{Risk Premium}$

This rate actually has several names (the discount rate, the cost of capital, the required rate of return, and the interest rate) and several possible abbreviations (r, i, k, and y), but all these refer to the same rate.1

We will leave further discussion of this rate for later in the text. For now, in order to understand the mechanics of time value calculations, we will just assume that the appropriate discount rate is given to us.

The Calculation Process

Time value of money calculations consist of moving money through time to find its value at different points on the timeline. There is a unique pair of terms, depicted in Figure 1.1, that we use when talking about time value of money calculations. Moving a sum of money further into the future (from left to right on the timeline) is known as compounding (i.e., calculating the future value of a sum). The opposite, moving a sum from the future back toward the present (or right to left on the timeline), is called discounting (i.e., calculating the present value of a sum). In finance jargon, we usually refer to the rate at which we move money through time as the discount rate, regardless of whether we are compounding or discounting.

The most intuitive way to do time value of money calculations is on a spreadsheet. Alternatively, you can use a financial calculator, though you will need a lot of practice to eliminate mistakes. While both of these methods make time value of money calculations quick and easy, it is still important to understand the underlying principles, formulas, and mathematics.