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 = 30,000, r = 5, t = 20, m = 4
To find the periodic payment R required to amortize a loan, we can use the formula for the present value of an ordinary annuity:
R = P * (r / 100 / m) * (1 + r / 100 / m)^(mt) / ((1 + r / 100 / m)^(mt) - 1)
In this formula:
- P is the principal amount (30,000 dollars)
- r is the annual interest rate (5%)
- t is the number of years (20 years)
- m is the number of times interest is compounded per year (4 times)
Let's calculate the periodic payment R:
Step 1: Convert the annual interest rate to decimal format: r / 100 = 5 / 100 = 0.05
Step 2: Convert the interest rate per period: r / 100 / m = 0.05 / 4 = 0.0125
Step 3: Calculate the factor (1 + r / 100 / m)^(mt): (1 + 0.0125)^(4 * 20) ≈ 1.183526676
Step 4: Calculate the periodic payment R:
R = 30,000 * 0.0125 * 1.183526676 / (1.183526676 - 1)
R ≈ 318.75
Therefore, the periodic payment R required to amortize the loan is approximately 318.75 dollars.