# 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.

2

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Description: This is an example of compound interest formula. This document is useful for conducting compound interest formula.
Richard Cataman