Question 1116656: Find the rate of change in the balance after 5 years if $4,000 is invested into an account that earns 3.5% compounded...
A)monthly
B)continuously
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! the discrete compounding formula is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.
the annual interest rate is 3.5% per year / 100 = .035 per year.
when the annual interest rate is compounded monthly, then:
the interest rate per time period = .035/12.
the number of time periods = 12 * the number of years.
since the number of years is 5, then the number of time periods is 12 * 5 = 60
your discrete compounding formula becomes:
f = 4000 * (1 + .035/12) ^ 60
this gets you a future value of 4763.771317.
this future value is equal to 4763.771317 / 4000 = 1.190942829 times the original value of 4000.
this would be an increase in the value of .190942829 = 19.0942829%.
the continuous compounding formula is f = p * e ^ (r * n)
f is the future value
p is the present value
e is the scientific constant of 2.718281828...
r is the interest rate per time period
n is the number of time periods.
unlike in the discrete compounding formula, r and n can be for any time period as long as they are consistent with each other.
if r is an annual interest rate, then n has to be number of years.
if r is a monthly interest rate, then n has to be number of months.
an example of this will be shown below as i go through the continuous compounding formula for you.
you are given that the interest rate per year = 3.5% which is equivalent to an interest rate of .035 per year.
since p = 4000 and n = 5 years, the formula becomes f = 4000 * e ^ (.035 * 5).
this results in f = 4764.984866.
f/p = 4764.984866 / 4000 = 1.191246217.
this results in an increase of .191246217 * 100 = 19.1246217%.
if your interest rate was monthly, then r = .035 / 12 and n = 5 * 12.
the formula would then become f = 4000 * e ^ (.035 / 12 * 5 * 12).
the result would be f = 4764.984866 which is the same as we got when using annual interest rate and number of years.
this is because the interest rate divided by the number of compounding periods per year and the number of years times the number of compounding periods per year cancels out.
1/12 * 12 = 1, so it doesn't matter whether you use annual interest rate and number of years of annual interest rate divided by the compounding periods per year and number of years times number of compounding periods per year.
the answer will be the same.
this is unlike discrete compounding, where it does make a difference depending on the number of compounding periods per year.
note that continuous compounding is the best that you can do, i.e. it gives you the highest overall rate of increase.
in fact, the definition of continuous compounding is the limit of (1 + r/c) ^ (n * c) as c approaches infinity.
in this formula, r is the interest rate per year, n is the number of years, c is the number of compounding periods per year.
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