Find the payment necessary to amortize a 4% loan of $1700 compounded quarterly, with 19 quarterly payments.
PV = 1700
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 calculating the monthly mortgage payment, which is:
Payment = P × (r × (1 + r)^n) / ((1 + r)^n - 1)
Where:
P = Principal amount (loan amount)
r = Monthly interest rate (annual interest rate / number of compounding periods per year)
n = Total number of payments (number of compounding periods per year × number of years)
In this case, the loan amount (P) is $1700, the annual interest rate is 4%, and there are 19 quarterly payments.
First, we need to calculate the monthly interest rate (r):
r = 4% / 4 = 0.01 (since it is compounded quarterly)
Next, we need to calculate the total number of payments (n):
n = 4 compounding periods per year × (19/4) years = 19
Now, we can substitute these values into the formula to calculate the payment:
Payment = 1700 × (0.01 × (1 + 0.01)^19) / ((1 + 0.01)^19 - 1)
Simplifying the equation, we get:
Payment = 1700 × (0.01 × 1.010050167)^19) / ((1.010050167)^19 - 1)
Payment = 1700 × (0.010150124165) / (1.19313944005 - 1)
Payment = 1700 × (0.010150124165) / 0.19313944005
Payment = $88.28
Therefore, the payment necessary to amortize the loan of $1700 compounded quarterly, with 19 quarterly payments, is $88.28.
To find the payment necessary to amortize a loan, we can use the formula for the present value of an annuity. The formula is:
Payment = (Loan amount / Present value factor)
To calculate the present value factor, we can use the formula:
Present value factor = (1 - (1 + interest rate/number of periods)^(-number of periods))
Let's break down the given information:
Loan amount (P) = $1700
Interest rate (r) = 4% or 0.04 (in decimal)
Number of periods (n) = 19 (since it's compounded quarterly)
Payment (A) = ? (what we're trying to find)
First, let's calculate the present value factor:
Present value factor = (1 - (1 + r/n)^(-n))
= (1 - (1 + 0.04/4)^(-4 * 19))
Now, we substitute the values into the formula:
Payment = (P / Present value factor)
= ($1700 / Present value factor)
Calculating the present value factor will give us the value we substitute into the formula to find the payment. So, if you input all the values into a scientific calculator or a spreadsheet, you should get the payment necessary to amortize the loan of $1700.