Suppose now that we have an annual interest rate r and that we would like to compound
(or pay) interest n times per year. Then we would define the interest rate for the
compound period to be
After one period the amount would be A where A1 represents the
accumulated amount after one period and P is the principal amount. For the second
compound period, the ‘principal’ is now A1 ;i.e.,
A A( i)
so that, in general, we have
A A1( )
n n 1 i
P i n11 i
Note here that the n is still the number of times the interest is compounded in a year so
that A is the accumulated amount after one year.
For the second year of the investment, let’s note that An is the amount of the
principal. Thus, if we invest this new principal with the same interest rate and compound
periods for another year, our new total is
A A( i)n
P1 i)n i
This last equation tells us that if a principal
is invested for 2 years at an annual rate of interest r compounded n times per year, then
the total is A P . In general with the aforementioned parameters in mind and
letting the investment last t years, the accumulated amount A of the investment is given
by the formula
A P ,
which is the formula for compound interest.
(A) Find the account balance if $1000 is invested for 5 years at an annual rate of 3%
(B) How long will it take for the account to double?
Solution: (A) Note that
A P 1 i
(B) We seek t such that
ln(2) 12 ln(
or 23 years.
In the compound interest formula, A represents the account total after term has ended. For
convenience, let’s change the letter from A to F so that we keep the word ‘future’ in
mind. It follows then that the future value F of an investment is given by the equation
F P )t .
Example 2.2: How much money must be deposited so that the future value of an account
totals $3000 if the annual rate of interest is 5% and the compound period is monthly for a
term of 5 years?
0 (. 0
0 1 4 0
Solution: We seek P such that 3 0 P0 ) (after substituting for
variables and simplifying) so that
Example 2.2 suggests that, in general, the present value P is given by the equation,