# The interest rate

THE INTEREST RATE

Any person with common financial sense prefers \$1000 today rather than \$1,000 ten years from today.

Commonsense tells us to take the \$1000 today because we recognize that there is a time value to money. The immediate receipt of \$1000 provides us with the opportunity to put our money to work and earn interest. In a world in which all cash flows are certain, the rate of interest can be used to express the time value of money.

The rate of interest will allow us to adjust the value of cash flows, when ever they occur, to particular point in time. Given this ability, we will be able to answer more difficult questions, such as: which should you prefer \$1000 today or \$2000 ten years from today? To answer this question, it will be necessary to position time adjusted cash flows at a single point in time so that a fair comparison can be made.

If we allow for uncertainty surrounding cash flows to enter into our analysis, it will be necessary to add a risk premium to the interest rate as compensation for uncertainty. But for now, our focus is on the time value of money and the ways in which the rate of interest can be used to adjust the value of cash flows to a single point in time.

Most financial decisions, personal as well as business, involve time value of money considerations. We have already learned that the objectives of management should be to maximize shareholder wealth and this depends, in part, on the timing of cash flows.

One important application of the concepts stressed in this article will be to value a stream of cash flows. Indeed, much of the development of interest, cash flow depends on the understanding of time value to money. You will never really understand finance until you understand the time value of money.

The discussion that follows in this article cannot avoid being mathematical in nature. We focus on only a handful of formulas so that one can easily grasp the fundamentals. We start with a discussion of simple interest and use this as a springboard to develop the concept of compound interest. Also, to observe easily the effect of compound interest, most of the examples in this article assume an 8 % annual interest rate.

Simple Interest

Simple interest is interest that is paid (earned) on only the original amount, or principal, borrowed (lent). The dollar amount of simple interest is a function of three variables: the original amount borrowed (lent), or principal; the interest rate per time period; and the number of time periods for which the principal is borrowed (lent) The formula for calculating simple interest is

SI = Po(i)(n) —————– (1)

Where SI= simple interest in dollars
Po= principal, or original amount borrowed (lent) at time period 0.
i=interest rate per time period
n= number of time periods.

For example, assume that you deposit \$100 in a savings account paying 8% Simple Interest and keep it there for 10 years. At the end of 10 years, the amount of interest accumulated is determined as follows:

\$100(.08) (10) = \$80

To solve for the future (also known as the terminal value) of the account at the end of 10 years (FV10), we add the interest earned on the principal only to the original amount invested. Therefore,

FV10 = \$100 + [\$100(.08)(10)] = \$180

For any simple interest rate, the future value of an account at the end of â€˜nâ€™ periods is

FV n= P0 + SI = P0 + P0 (i) (n)

Or, equivalently,

FV n= Po [1+ (i) (n)] ———– (2)

Sometimes we need to proceed in the opposite direction. That is, we know the future value of a deposit at â€˜Iâ€™ percent for â€˜nâ€™ years, but we donâ€™t know the principal originally invested the accountâ€™s present value

(PV0 = P0 ).
A rearrangement of Eq (2), however, is all that is needed

PV 0 =P0 = FVn / [1 + (i) (n)] ——— (3)

The above makes us familiar with the mechanics of simple interest it is perhaps a bit cruel to point out that most situations in finance involving the time value of money do not rely on simple interest at all.

Instead, compound interest is the norm; however, an understanding of simple interest will help you appreciate (and understand) compound interest all the more.

Compound Interest

The distinction between simple and compound interest can best be seen by example Table below illustrates the rather dramatic effect that compound interest has on an investmentâ€™s value over time when compared to the effect of simple interest. From the table it is clear to see why some people have called the compound interest the greatest of human inventions.

Years—At simple interest—-At compound interest
–2———–\$ 1.16——————\$ 1.17
–20———–2.60——————-4.66
–200———–17.00—————-4,838,949.59

The notion of compound interest is crucial too understanding the mathematics of finance. The term itself merely implies that interest paid (earned) on a loan (an investment) is periodically added to the principal. As a result, interest is earned on interest as well as the initial principal. It is this interest-on-interest, or compounding effect that accounts for the dramatic difference between simple and compound interest.