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Compound Interest Formula

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					                      Sec 5.3 Compound Interest


Compound Interest Formula
                                          r mt
                            A = P (1 +      )
                                          m
where
     A = Accumulated amount at the end of t years.

     P = Principal

     r = Nominal interest rate per year

     m = Number of conversion periods per year

     t = Term (number of years)
Present Value Formula for Compound Interest
                                         r −mt
                            P = A(1 +      )
                                         m
Continuous Compound Interest Formula

                                A = P ert

where
     P = Principal

     r = Annual interest rate compounded continuously

     t = Time in years

     A = Accumulated amount at the end of t years
Present Value Formula for Continuous Compound Interest

                               P = Ae−rt

Example 1 Find the accumulated amount A if the principal P=$2500,
interest rate r=7%, after t=10 years, and compounded semiannually.
Solution. Compounding semiannually means that m=2,
                 r mt           0.07 (2)(7)
    A = P (1 +     ) = 2500(1 +     )       = 2500(1.035)14 = 4046.7
                 m                2

                                    1
Example 2 Find the interest rate needed for an investment of $5000 to
grow to an amount of $7500 in 3 yr if interest is compounded monthly.
Solution. Here P=5000, A=7500, t=3, m=12, from the formula
                                              r (3)(12)
                         7500 = 5000(1 +        )
                                             12
Solve this equation,
                                         r        3
                            36 ln(1 +      ) = ln
                                        12        2
then
                                     r    ln 3
                                             2
                               1+      = e 36
                                    12
then
                                            ln 3
                                               2
                            r = 12(1 − e     36    )=
Example 3 Find the interest rate needed for an investment of $4000 to
double in 5 yr if interest is compounded continuously.
 Here P=4000, A=(2)(4000)=8000, t=5, from the formula

                              8000 = 4000e5r

Then
                                  5r = ln 2
then
                                       ln 2
                                  r=
                                        5
Example 4 How long will it take an investment of $8000 to double if the
investment earns interest at the rate of 8% compounded continuously?
Solution. Here A=(2)(8000)=16000, and P=8000, and r=0.08, from the
formula
                             16000 = 8000e0.08t
then
                                        ln 2
                                  t=
                                       0.08
    In real application, we have options to choose the best strategy. There-
fore, we have to compare the results from different plans or different formula.
Either we want the shortest time or the largest accumulated compound.




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Richard Cataman Richard Cataman
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