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Are you indifferent between receiving $1,000 today and receiving $1,000 one year from today? If your intuition prefers receiving the funds today, rather than one year from today, then your intuition recognizes the time value of money. Owners of cash can permit borrowers to rent the use of their cash. Interest is payment for the use of cash.

Expenditures for an investment most often precede the receipts produced by that investment. Cash received later has less value than cash received sooner. The difference in timing affects whether making an investment will earn a profit. Amounts of cash received at different times have different values. We use interest calculations to make valid comparisons among amounts of cash paid or received at different times.

CONCEPTS

Businesses typically state interest cost as a percentage of the amount borrowed per unit of time. Examples are 12 percent per year and 1 percent per month. When the statement of interest cost includes no time period, then the rate applies to a year; thus "interest at the rate of 12 percent" means 12 percent per year.

The amount borrowed or loaned is the principal. Compound interest means that the amount of interest earned during a period increases the principal, which is then larger for the next interest period.

If you deposit $1,000 in a savings account that pays compound interest at the rate of 6 percent per year, you will earn $60 by the end of one year. If you do not withdraw the $60, then $1,060 will earn interest during the second year. During the second year your principal of $1,060 will earn $63.60 interest; $60 on the initial deposit of $1,000 and $3.60 on the $60 earned the first year. By the end of the second year, you will have $1,123.60. Compounded annually at 8 percent, cash doubles itself in nine years. If a twenty-five-year old invests $2,000 each year which earns 8 percent a year, the retirement fund will grow to more than $425,000 by the time that person reaches age sixty-five.

When only the original principal earns interest during the entire life of the loan, the interest due at the time the borrower repays the loan is simple interest. Simple interest calculations ignore interest on previously earned interest. Nearly all economic calculations, however, involve compound interest.

Problems involving the time value of money generally fall into two groups:

1. We want to know the future value of cash invested or loaned today.
2. We want to know the present value, or today's value, of cash to be received or paid at later dates.

FUTURE VALUE

If you invest $1 today at 12 percent compounded annually, it will grow to $1.12000 at the end of one year, $1.25440 at the end of two years, $1.40493 at the end of three years, and so on, according to the formula:

Fn; = P (1 + r)_n

where

F_n = accumulation or future value

P = one-time investment today

r = interest rate per period

n = number of periods from today

The amount F_n is the future value of the present payment, P, compounded at r percent per period for n periods.

Example. How much will $2,000 deposited today at 8 percent compounded annually be worth 40 years from now?

$2,000 will grow to $2,000 × (1.08)⁴⁰ = $2000 × 21.72452 = $434,490 *

*(While you can compute 1.08 can be raised to the 40th power manually, future value tables, calculation, and computers can remove the tedium of such computations.)

PRESENT VALUE

Now, consider how much principal, P, you must invest today in order to have a specified amount, F_n, at the end of n periods. You know the future amount, F_n, the interest rate, r, and the number of periods, n; you want to find P. In order to have $1 one year from today when deposits earn 8 percent, you must invest P of $.92593 today. That is, F₁ = P (1.08)¹ "or" $1 = $.92593 × 1.08.

The number (1 + r) - _n [= 1/(1 + r)ⁿ] equals the present value of $1 to be received after n periods when interest accrues at r percent per period. The discounted present value of $1 to be received n periods in the future is (1 + r) - n when the discount rate is r percent per period for n periods.

Example What is the present value of $1 due 10 years from now if the interest rate (equivalently, the discount rate) r is 8 percent per year? (1 + .08)-¹⁰ × $1 = $.46319 *

*(Present value tables and computers simplify such a calculation)

CHANGING THE COMPOUNDING PERIOD: NOMINAL AND EFFECTIVE RATES

"Twelve percent, compounded annually" states the price for a loan; this means that interest increases principal once a year at the rate of 12 percent. Often, however, the price for a loan states that compounding is to take place more than once a year. A savings bank may advertise that it pays 6 percent, compounded quarterly. This means that at the end of each quarter the bank credits savings accounts with interest calculated at the rate 1.5 percent (= 6%/4).

$10,000 invested today at 12 percent, compounded annually, grows to a future value one year later of $11,200. If the rate of interest is 12 percent compounded semiannually, the bank adds 6 percent interest to the principal every six months. At the end of the first six months, $10,000 will have grown to $10,600; that amount will grow to $10,600 × 1.06 = $11,236 by the end of the year. Notice that 12 percent compounded semiannually is equivalent to 12.36 percent compounded annually. At 12 percent compounded monthly, $1 will grow to $1 × (1.01)12 = $1.12683 and $10,000 will grow to $11,268. Thus, 12 percent compounded monthly provides the same ending amount as 12.68 percent compounded annually. Common terminology would say that 12percent compounded monthly has an effective rate of 12.68 percent compounded annually or is equivalent to 12.68 percent compounded annually. If a nominal rate, r, compounds m times per year, the effective rate equals (1 + r/m)^m - 1.