Sec 5.3 Compound Interest
Compound Interest Formula
A = P (1 + )
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
P = A(1 + )
Continuous Compound Interest Formula
A = P ert
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
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
7500 = 5000(1 + )
Solve this equation,
36 ln(1 + ) = ln
r ln 3
1+ = e 36
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
5r = ln 2
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
16000 = 8000e0.08t
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.