Machinery's Handbook, 31st Edition
INTEREST 145 Example: Determine the principal needed for an account to have a future value of $25,000 ( F ) in 10 years ( n ), given that interest is compounded quarterly (four times a year; so m = 4) at an annual rate of 2.5% ( i = 0.025). Solution: P is sought, so the formula is: P F i m mn = + − 1 ; ( ) P = + = = − × 25000 1 0025 4 25000 0 7794 19 485.17 4 10 . ( . ) $ , ( ) To find i when the other parameters of the loan or investment are known, use the formulas: At compound interest: i F P mn = − 1 Solving for Time (n): Solving the future value formulas for n requires logarithms. The process for compound interest is shown (for properties of logarithms, see page 36):
i m ( ) +
F P ( )
i m ( )
F P ( ) ( ) i m = + 1 log
i m ( )
mn
mn
m
n
F P
F P
mn
n → =
= + 1
= + 1
=
→
→
log
→
log
log 1
F P ( ) ( ) ( ) ( ) NOTE: From logarithm properties, log ( a / b ) = log a – log b . Time to Increase Investment (n): A related approach is taken when calculating time to reach a specific total value: Compound interest: n i m m = + 1 log log n F P ( ) mn i m ( ) n → = F P n i m → = + + log log log log 1 1 F P ( ) ( ) Example: Determine the time required to double $500, when interest is compounded monthly at a rate of 5%. Solution: Here, i = 0.05, m = 12, P = $500, and F = 2 P = $1000. n = + ( ) = = log log . log log . 1000 500 12 1 005 12 2 12 10042 13.9 years ( ) ( ) Doubling Time for an Investment: The time needed to double an investment is the same for any principal, since the variables F and P drop out. Compound interest: n n i n double = + log log 2 1 ( ) Nominal versus Effective Interest Rates.— Deposits in savings banks, automobile loans, interest on bonds, and many other transactions of this type involve computation of interest due and payable more often than once a year. For such instances, there is a dif- ference between the nominal annual interest rate (the stated rate) for the cost of borrowed money and the effective annual interest rate (the amount that is actually accrued). The formula for calculating the effective interest rate (EIR) is: EIR = (1 + i / m ) m – 1 The nominal interest rate is also called the “APR,” or the annual percentage rate. APR is quoted without compounding frequency per year. If compounding is done only at the end of the year, the APR is the same as the effective rate. If compounding happens more than once in a year, then the effective rate will be greater than the APR. In this case, it is essential the frequency of compounding be stated when APR is quoted. For example, simply saying APR is 10% is not sufficient. Rather, it must be stated that the APR is 10% compounded yearly, for example, or that it is compounded monthly. n i m = + log log 1
i m ( )
mn
F P
= + 1
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