Find the periodic payment R required to amortize a loan of P dollars over t years with interest charged at the rate of r%/year compounded m times a year. (Round your answer to the nearest cent.)
P = 40,000, r = 3, t = 13, m = 12
I'm sure you have this formula. So plug in your numbers.
To find the periodic payment R required to amortize a loan, we can use the formula for the monthly payment on an amortized loan:
R = (P * r / (m * 100)) / (1 - (1 + r / (m * 100))^(-m * t))
Let's plug in the values given:
P = 40,000 (principal)
r = 3 (interest rate)
t = 13 (duration in years)
m = 12 (number of compounding periods per year)
First, let's convert the interest rate to a decimal:
r = 3% = 3/100 = 0.03
Now, substitute the values into the formula:
R = (40,000 * 0.03 / (12 * 100)) / (1 - (1 + 0.03 / (12 * 100))^(-12 * 13))
Simplifying the equation inside the parenthesis:
R = (1,200 / (12 * 100)) / (1 - (1 + 0.00025)^-156)
R = (1,200 / 1,200) / (1 - (1.00025)^-156)
R = 1 / (1 - (1.00025)^-156)
R ≈ 1 / (1 - 0.442437)
R ≈ 1 / 0.557563
R ≈ 1.79189
Therefore, the periodic payment required to amortize the loan is approximately $1.79.