Find the least amount that could be deposited in a bank account today at 5% compounded quarterly to allow $650 withdrawals at the end of each quarter for 4 years.
present value = 650(1 - 1.0125^-16)/.0125
= ....
To find the least amount that could be deposited in a bank account today, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the account (in this case, $0 at the end of 4 years)
P = the principal amount (the amount we are trying to find)
r = the annual interest rate (5% expressed as 0.05)
n = the number of times the interest is compounded per year (quarterly, so 4)
t = the number of years (4)
Since we want to calculate the minimum amount that needs to be deposited initially, we can assume that the future value A is 0 and solve for P:
0 = P(1 + 0.05/4)^(4*4)
Simplifying the equation:
0 = P(1.0125)^16
Dividing both sides by (1.0125)^16:
0 / (1.0125)^16 = P
P ≈ $528.65
So, the least amount that could be deposited in the bank account today at a 5% interest rate compounded quarterly to allow $650 withdrawals at the end of each quarter for 4 years is approximately $528.65.
To find the least amount that could be deposited in a bank account today, we need to calculate the present value of the quarterly withdrawals over 4 years at a 5% interest rate compounded quarterly.
The formula to calculate the present value of a series of future cash flows is:
PV = CF / (1 + r)^n
Where:
PV = Present Value (amount to be deposited today)
CF = Cash Flow (withdrawal amount at the end of each quarter) = $650
r = Interest rate per compounding period (quarterly) = 5% = 0.05
n = Number of compounding periods (4 years with quarterly compounding) = 4 * 4 = 16
Substituting the values into the formula:
PV = $650 / (1 + 0.05)^16
Now, let's calculate this using a calculator:
PV = $650 / (1.05)^16
PV = $650 / 1.87354
PV ≈ $346.80
Therefore, the least amount that could be deposited in a bank account today, at a 5% interest rate compounded quarterly, to allow $650 withdrawals at the end of each quarter for 4 years is approximately $346.80.