The monthly payment that amortizes a loan of A dollars in t...

The monthly payment that amortizes a loan of $A$ dollars in $t$ years when the interest rate is $r /$ year, compounded monthly, is given by $$P=f(A, r, t)=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}$$ a. What is the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 30 years with an interest rate of $4 \% /$ year? An interest rate of $6 \% /$ year? b. Find the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 20 years with an interest rate of $6 \% /$ year.

The monthly payment that amortizes a loan of $A$ dollars in $t$ years when the interest rate is $r /$ year, compounded monthly, is given by $$P=f(A, r, t)=\frac{A r}{12\left[1-\left(1+\frac{r}{12}\right)^{-12 t}\right]}$$
a. What is the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 30 years with an interest rate of $4 \% /$ year? An interest rate of $6 \% /$ year?
b. Find the monthly payment for a home mortgage of $\ 300,000$ that will be amortized over 20 years with an interest rate of $6 \% /$ year.

Key Concepts

Loan Amortization

Loan amortization refers to the process of gradually paying off a debt over time through regular payments. Each payment covers both the interest accruing on the outstanding balance and a portion of the principal. Over the life of the loan, the interest component decreases while the principal component increases, ensuring the loan is completely repaid by the end of the term.

Compound Interest

Compound interest is the addition of interest to the principal of a loan, where interest is earned on both the initial principal and the accumulated interest from previous periods. This concept is central to calculating payments because it explains how the loan balance grows over time if no payments are made or how the interest reduces as the principal decreases.

Periodic Interest Rate

When interest is compounded more frequently than annually, the annual interest rate must be converted to a periodic rate by dividing by the number of compounding periods per year. This adjusted rate is used in calculations to determine the correct amount of interest for each payment period.

Annuity Formula

An annuity represents a series of equal payments made at regular intervals. The mortgage payment formula is derived from the annuity formula, which computes the present value of a series of future payments. This allows one to solve for the fixed payment that will amortize the loan given the interest rate, loan term, and principal amount.

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