If an amount of money
is
invested so that compound interest is accrued at the rate r% per time
period, then after
time
periods the amount of money will have grown to an amount
is
invested so that compound interest is accrued at the rate r% per time
period, then aftertime
periods the amount of money will have grown to an amount
If however, the interest is
compounded more regularly, then something a little bit strange
happens. Suppose £1000 is invested at 12% per annum. If interested
is compounded annually then at the end of a year, the original £1000
will have grown to £1120. If however, it is compounded monthly, then
the monthly rate of interest will be 12/12 =1% and after 1 year the
original £1000 will have grown to
In fact if the year is divided
intotime
periods, so that interest is compounded n times a year, the interest
per time period is
and
the amount of money will have grown to
time
periods, so that interest is compounded n times a year, the interest
per time period isand
the amount of money will have grown to
The table below shows the
investment after 1 year for various values of n.
|
|
 |
|
10
|
1126.691779
|
|
100
|
1127.415743
|
|
1000
|
1127.488731
|
|
10000
|
1127.495196
|
|
100000
|
1127.495975
|
Astends
to infinity, this expression tends to a limit
tends
to infinity, this expression tends to a limit
We can generalise this reasoning,
so that if annual interest of
is
compounded continuously on an investment of
at
the end of a year the investment will have grown to
and at the end ofyears
the principal will have grown to
is
compounded continuously on an investment ofat
the end of a year the investment will have grown to
and at the end ofyears
the principal will have grown to
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