Question 1096036: Bob can invest $3000.00 in a continuously compounded account or he can invest his $3000.00 in an account compounded monthly. Both accounts pay 2 1/4% interest.
a) which account would pay more ? Why ?
b) how much more will Bob earn ?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the continuous compounding should pay more.
that's because continuous compounding gives you the maximum number of compounding periods per year.
the formula for discrete compounding is:
f = p * (1+r)^n
r is the interest rate per time period.
n is the number of time periods.
the formula for continuous compounding is f = p * e^(rn)
r is the interest rate per time period.
n is the number of time periods.
you did not specify the number of years, so any number of years is applicable.
you did not specify the number of discrete compounding periods per year so we'll assume 1.
our time periods is therefore assumed to be in years.
therefore the interest rate for discrete time periods is compounding once a year and the interest rate per time period is 2.225 / 100 = .0225
similarly, the time periods for continuous time periods will use the same rate and assume years.
you get f = 3000 * (1.0225)^n for discrete compounding.
you get f = 3000 * e^(.0225*n) for continuous compounding.
after 100 years, your results will be:
f = 3000 * (1.0225)^100 = 27762.14
f = 3000 * e^(.0225*100) = 28463.20751
continuous compounding gives you more.
after 1 year, your results are as follows.
f = 3000 * 1.0225)^1 = 3067.5
f = 3000 * e^(.0225*1) = 3068.265102
continuous compounding gives you an infinite number of compounding periods per year.
to simulate, you can assume a very large number of compounding periods per year.
assume 1000 compounding periods per year.
your annual rate of .0225 becomes a time period rate of .0225/1000.
your number of years becomes 1000 * number of years = 1000 time periods.
for 1 year, you will get:
3000 * (1 + .0225/1000) ^ (1 * 1000) = 3068.264326
that's pretty close to the continuous compounding result of 3068.26512.
it's not quite as much because even 1000 compounding periods per year isn't as many compounding periods per year as continuous.
so, bottom line:
continuous will give you more because the number of compounding periods per year is infinite.
compounding gives you more because you are earning interest on your interest.
the more compounding periods per year, the more you earn for the same annual rate.
with monthly compounding your future value after 1 year will becomes:
3000 * (1 + .0225/12) ^ (1 * 12) = 3068.200463.
that's more than annual compounding but less than continuous compounding.
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