Find the least amount that could be deposited in a bank account today at 10% compounded quarterly to allow $650 withdrawals at the end of each quarter for 8 years?
To find the least amount that could be deposited in a bank account today, we need to compute the present value of the annuity that will allow for $650 withdrawals at the end of each quarter for 8 years, with a 10% interest rate compounded quarterly.
The formula to calculate the present value of an annuity is:
PV = PMT x (1 - (1 + r)^-n) / r
Where:
PV = Present Value (the amount to be deposited today)
PMT = Payment amount at the end of each period ($650)
r = Interest rate per period (10% / 4 = 0.1 / 4 = 0.025)
n = Number of periods (8 years x 4 quarters = 32 quarters)
Let's substitute the values into the formula and calculate the present value:
PV = 650 x (1 - (1 + 0.025)^-32) / 0.025
PV = 650 x (1 - 0.4531) / 0.025
PV = 650 x 0.5469 / 0.025
PV ≈ 14,108.8
Therefore, the least amount that could be deposited in the bank account today is approximately $14,108.8.
To find the least amount that could be deposited in a bank account, we need to use the formula for compound interest:
A = P * (1 + r/n)^(nt)
Where:
A = the future value of the bank account
P = the principal amount (initial deposit)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year
t = the number of years
In this case, we need to find the principal amount (P), given the future value (A), the interest rate (r), the number of times compounded per year (n), and the number of years (t). We want to find the minimum value of P that allows for $650 withdrawals at the end of each quarter for 8 years.
First, let's determine the values for the equation:
A = $650 (withdrawal amount) * 4 (quarters in a year) * 8 (number of years) = $20,800
r = 10% (annual interest rate) = 0.10
n = 4 (compounded quarterly)
t = 8 (years)
Now we can plug these values into the equation and solve for P:
$20,800 = P * (1 + 0.10/4)^(4*8)
Simplifying the equation:
$20,800 = P * (1.025)^32
To solve for P, divide both sides of the equation by (1.025)^32:
P = $20,800 / (1.025)^32
Using a calculator, we can evaluate this:
P ≈ $20,800 / 2.2083097 ≈ $9,413.58 (rounded to the nearest cent)
Therefore, the least amount that could be deposited in the bank account today at 10% compounded quarterly, allowing for $650 withdrawals at the end of each quarter for 8 years, is approximately $9,413.58.