Find the payment necessary to amortize a 4% loan of $1900 compounded quarterly, with 19 quarterly payments.
PV = 1900
i = .04/4 = .01
n = 19
paym = ?
PV = paym( 1 - (1+i)^-n ) / i
sub in the values, and solve for paym
To find the payment necessary to amortize a loan, we can use the formula for the present value of an annuity.
The formula for calculating the present value of an annuity is:
PV = PMT * [1 - (1+r)^(-n)] / r
Where:
PV = Present value or loan amount
PMT = Payment amount
r = Interest rate per compounding period
n = Number of compounding periods
In this case, the loan amount is $1900, the interest rate is 4% per year, compounded quarterly, and there are 19 quarterly payments.
First, we need to find the interest rate per compounding period (quarterly in this case). We can do this by dividing the annual interest rate by the number of compounding periods in a year:
r = 4% / 4 = 1%
Next, we can substitute the values into the formula:
PV = PMT * [1 - (1+r)^(-n)] / r
1900 = PMT * [1 - (1+0.01)^(-19)] / 0.01
Now, we can solve the equation for PMT, which is the payment amount necessary to amortize the loan:
PMT = 1900 * 0.01 / [1 - (1+0.01)^(-19)]
Calculating this expression will give us the payment amount necessary to amortize the loan.