Suppose a home buyer secures a bank loan of A dollars to...

Suppose a home buyer secures a bank loan of $A$ dollars to purchase a house. If the interest rate charged is $r /$ year compounded monthly and the loan is to be amortized in $t$ years, then the principal repayment at the end of $i$ months is given by $$\begin{aligned} B &=f(A, r, t, i) \\ &=A\left[\frac{\left(1+\frac{r}{12}\right)^{i}-1}{\left(1+\frac{r}{12}\right)^{12 t}-1}\right] \quad(0 \leq i \leq 12 t) \end{aligned}$$ Suppose the Blakelys borrow a sum of $\ 280,000$ from a bank to help finance the purchase of a house and the bank charges interest at a rate of $6 \% /$ year. If the Blakelys agree to repay the loan in equal installments over 30 years, how much will they owe the bank after the 60th payment (5 years)? The 240th payment (20 years)?

Suppose a home buyer secures a bank loan of $A$ dollars to purchase a house. If the interest rate charged is $r /$ year compounded monthly and the loan is to be amortized in $t$ years, then the principal repayment at the end of
$i$ months is given by $$\begin{aligned}
B &=f(A, r, t, i) \\
&=A\left[\frac{\left(1+\frac{r}{12}\right)^{i}-1}{\left(1+\frac{r}{12}\right)^{12 t}-1}\right] \quad(0 \leq i \leq 12 t)
\end{aligned}$$
Suppose the Blakelys borrow a sum of $\ 280,000$ from a bank to help finance the purchase of a house and the bank charges interest at a rate of $6 \% /$ year. If the Blakelys agree to repay the loan in equal installments over 30 years, how much will they owe the bank after the 60th payment (5 years)? The 240th payment (20 years)?

Key Concepts

Compound Interest

Compound interest is the process by which interest is earned not only on the initial principal amount but also on the accumulated interest from previous periods. When interest is compounded monthly, as in this case, the interest rate is divided by the number of periods per year and interest is added at the end of each month, resulting in exponential growth of the amount owed if not repaid.

Amortization of Loans

Amortization is the method of paying off a debt over time through regular, scheduled payments that cover both the interest due and a portion of the principal. With each payment, the portion that goes toward the principal increases while the interest portion decreases, ensuring that the entire loan is repaid over the predetermined term.

Outstanding Balance Calculation

The outstanding balance of a loan at any given point is determined by calculating the total amount of principal repaid up to that time and subtracting it from the original loan amount. In the context of compounded interest, formulas for amortization often involve exponential terms to accurately represent the impact of interest accrual over time.

Fixed Payment Loans

Fixed payment loans, such as many mortgages, require the borrower to make equal periodic payments throughout the life of the loan. These payments are computed using the principles of compound interest and amortization, ensuring that the loan is gradually paid down with each installment while covering both interest and principal repayment.

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