Machinery's Handbook, 31st Edition
144 INTEREST Simple Interest.— Over n years, the simple interest formulas are: Value of interest ( I ): I = Pni Future account value ( F ): F = P + I = P + Pni = P (1 + ni ) Example: $250 is lent for three years at 6 percent simple interest. P = 250, i = 6/100 = 0.06, n = 3. Then: F = 250 + (250 × 3 × 0.06 ) = 250[1 + (0.06 × 3)] = 250[1.18] = $295 Compound Interest.— If compound interest is calculated multiple times in a year, say m times a year, then the future value F of principal P over n years at annual rate i is found by the formula: F P i m mn = + 1 ( ) Example: Suppose $10,000 is borrowed from a bank as discussed above for 2 years at 10% APR or nominal interest rate; there are three ways the bank can calculate interest: Simple interest over 2 years: The borrower will pay F = P + Pni = 10,000 + (10,000 × 2 × 0.1) = 10,000 + 2,000 = $12,000. Compound interest over 2 years with a nominal interest rate of 10% compounded only at the end of the year: F = P [1 + ( i / m )] mn = 10,000 [1 + (0.10/1)] 1 × 2 = 10,000 × 1.1 2 = 10,000 × 1.21 = $12,100. This is the same the calculation at the beginning of the discussion. Compound interest over 2 years with a nominal interest rate of 10% compounded four times a year, or m = 4: F = P [1 + ( i / m )] mn = 10,000 [1 + (0.10/4)] 4 × 2 = 10,000 × 1.025 4 × 2 = 10,000 × 1.2184 = $12,184. Hence, the interest compounded multiple times ( m = 4) in a year gives more interest than if compounded only once a year. The effect of the four com- pounds is that the interest is higher by $84. Clearly, compounding yields more interest than simple interest. This example shows the effect of once a year compounding over a number of years, compared to a single calcula- tion at the end of the same number years. Determining Principal, Rate, or Time.—What principal should be invested to reach a known or desired future value, based on a given interest rate and time? At what annual rate should an investment be made when the other parameters are known? How much time will a principal take to reach a future value given an annual rate of interest? Solutions to these questions are found by transposing the main formula to isolate the unknown in question. Note: In the formulas below, the units for time are years. If time is given in months or any other unit, it must be converted to years before substituting it into the formula. Principal Value of an Investment or Loan (P): These days all commercial banks use compound interest only, so only the formulas and steps for calculating compound interest are shown: Simple interest: so F P ni = + ( 1 P F F ni ( 1 = = + − ), ) 1
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Example: Determine the initial deposit (principal) required for an account to reach $8,000 ( F ), in 10 years ( n ), at an annual interest rate of 4% ( i = 0.04), compounded monthly ( m = 12). Solution: Substituting known values into the formula: P F r n nt = + = = = ( ) − + − × − 1 8000 8000 1 003333 1 0 04 12 12 10 120 . ( . ) $5,366 ( )
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