SOLUTION: If you need $20,000 six years from now, what is the minimum amount of money you need to deposit into a bank account that pays 4% annual interest, compounded (give your answers to t

Question 1201055: If you need $20,000 six years from now, what is the minimum amount of money you need to deposit into a bank account that pays 4% annual interest, compounded (give your answers to the nearest cent):
(a) annually? $
(b) monthly? $
(c) daily (assuming 365 days in a year)? $
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
the formula for future value of an investmnt is f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interst rate per time period.
n is the number of time periods.

4%, compounded annually, is equal to an interest rate of 4/100 = .04.
add 1 to that to get a growth factor of 1.04 per year.
n = 6 years
p = 20,000
formula becomes f = 20,000 * 1.04 ^ 6 = 25306.38.

4$, compounded monthly is equal to an interest rate of 4/1200 = .0033333333 per month.
add 1 to that to get a growth factor of 1.0033333333 per month.
n = 6 years * 12 = 72 months
p = 20,000
formula becomes f = 20,000 * 1.0033333333 ^ 72 = 25414.84.

4%, compounded daily, is equal to an intnerest rate of 4/36500 = .0001095890411 per day.
add 1 to that to get a growth factor of 1.0001095890411
n = 6 years * 365 = 2190 days.
p = 20,000
formula becomes f = 20,000 * 1.0001095890411 ^ 2190 = 25424.65.

your solutions are:
future value is 25306.38 when the interest rate is compounded annually.
future value is 25414.84 when the interest rate is compounded monthly.
future value is 25424.65 when the interest rate is compounded daily.

the more times you compound the interest rate per year, the higher the future value will becomes.
the upper limit for the future value appears to be when you do continuous compounding.
that uses another formula, which is f = p * e ^ (r * n)
in your problem, this becomes f = 20,000 * e ^ (.04 * 6) = 25424.98.
daily compounding gets pretty close to that, but will never be as high.
the value of e is normaly shown as e = 2.718281828.
it is an irrational number that has an endless number of decimal places.
the value shown is rounded to the display of the calculator used.

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