Find the payment necessary to amortize a 4% loan of $900.00 compounded quarterly , with 11 quarterly payments?
To find the payment necessary to amortize a loan, we can use the formula for the amortization payment:
P = (r * PV) / (1 - (1 + r)^(-n))
Where:
P = Payment
PV = Present Value (loan amount)
r = Interest rate per period
n = Number of periods
In this case, the loan amount (PV) is $900.00, the interest rate per period (r) is 4% or 0.04 (dividing by 100 to convert it to a decimal), and the number of periods (n) is 11 quarters.
To find the interest rate per period, we need to divide the annual interest rate (4%) by the number of compounding periods per year (quarterly compounding, so 4 quarters in a year):
r = Annual Interest Rate / Number of Compounding Periods
= 4% / 4
= 1%
Now we can substitute the values into the formula:
P = (0.01 * $900) / (1 - (1 + 0.01)^(-11))
Calculating the exponent:
(1 + 0.01)^(-11) = 0.8873
Substituting the values:
P = (0.01 * $900) / (1 - 0.8873)
= $9.00 / 0.1127
= $79.89
Therefore, the payment necessary to amortize a 4%, $900.00 loan compounded quarterly with 11 quarterly payments would be $79.89.
To find the payment necessary to amortize a loan, we can use the formula for the present value of an annuity.
The formula for the present value of an annuity is:
P = C * [1 - (1 + r)^(-n)] / r,
Where:
P = present value of the annuity (loan amount)
C = periodic payment
r = interest rate per compounding period
n = total number of compounding periods (payments)
In this case,
P = $900.00 (loan amount)
r = 4% per year compounded quarterly = 4% / 4 = 1% per compounding period
n = 11 quarterly payments
Let's plug these values into the formula and calculate the periodic payment (C):
C = P * (r / [1 - (1 + r)^(-n)])
= $900.00 * (1% / [1 - (1 + 1%)^(-11)])
= $900.00 * (1% / [1 - (1.01)^(-11)])
= $900.00 * (0.01 / [1 - 0.90537])
= $900.00 * (0.01 / 0.09463)
= $900.00 * 0.10573
= $95.157
Therefore, the payment necessary to amortize the loan is approximately $95.16.