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The Tlme Value of Money
The lnterest Rate

Which would you prefer - $1,000 today or $1,000 ten years from today? Common sense tells us to take the $1,000 today because we recognize that there is a time value to money. The immediate receipt of $1,000 provides us with the opportuniql 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. As we will soon discover, the rate of interest will allow us to adjust the vaiue of cash flows, whenever they occur, to a particular point in time. Given this ability, we will be able to answer more difficult questions, such as: which should you prefer - $1,000 today or $2,000 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.

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Simpte lnterest

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 funciion of three variables: the originai amount borrowed (lent), or principal; the interest rate per time period; and the number of time periods for which the principal is borrowed (lentj. The foimula for calculating simple interest is

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Singte Amounts

The distinction between simple and compound interest can best be seen by example. Illustrates the rather dramatic effect that compound interest has on an investment's value over time when compared with the effect of simple interest. From the table it is clear to see why some people have called compound interest the greatest of human inventions. The notion of compound interest is crucial to 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. As we will see, the concept of compound interest can be used to solve a wide variety of problems in finance.

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Annuities

Ordinary Annuity. An annuity is a series of equal papnents or receipts occurring over a specified number of periods. In an ordinary annuity, payments or receipts occur at the end of each period. Figure 3.3 shows the cash-flow sequence for an ordinary annuity on a time line. Assume that Figure 3.3 represents your receiving $1,000 a year for three years. Now iet's further assume that you deposit each annual receipt in a savings aciount earning B percent compound annual interest. How much money will you have at the end of three years?

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Mixed Flows

Many time value of money problems that we face involve neither a single cash flow nor a single annuity. Instead, we may encounter a mixed (or uneven) pattern of cash flows.

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SemiannuaI and Other Compounding Periods

Future (or Compound) Vatue. Up to now, we have assumed that interest is paid annually. It is easiest to get a basic understanding of the time value of money with this assumption. Now, however, it is time to consider the relationship between future value and interest rates for different compounding periods. To begin, suppose that interest is paid semiannually. If you then deposit $100 in a savings account at a nominal, or stated,8 percent annual interest rate, the future value at the end of six months would be

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Continuous Compounding

In practice, interest is sometimes compounded continuously. Therefore it is useful to consider how this works. Recall that the general formula for solving for the future value at the end of year n, Eq. (3.17), is

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Effective AnnuaI lnterest Rate

Different investments may provide returns based on various compounding periods. If we want to compare alternative investments that have different compounding periods, we need to state their interest on some common, or standardized, basis. This leads us to make a distinction between nominal, or stated, interest and the effective annual interest rate. The effective annual interest rate is the interest rate compounded annually that provides the same annual interest as the nominal rate does when compounded m times per year.

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Amortizing a Loan

An important use of present value concepts is in determining the payments required for an installment-type loan. The distinguishing feature of this loan is that it is repaid in equal periodic payments that include both interest and principal. These payments can be made monthly, quarterly, semiannually, or annually. Installment payments are prevalent in mortgage loans, auto loans, consumer loans, and certain business loans.

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