Amortization is crucial in many real-world financial situations, including mortgages, car loans, and student loans.
The amortization process typically follows these steps:
As the loan progresses, the proportion of each payment going towards the principal increases, while the interest portion decreases.
Scenario: Sarah buys a house and takes out a $\$ 250,000$ mortgage at an annual interest rate of $\mathbf{4 \%}$, to be repaid over 30 years with monthly payments.
Using the loan payment formula: $$\text { Payment }=P \times \frac{r(1+r)^n}{(1+r)^n-1}$$ where: - $P=250,000$ (loan amount), - $r=\frac{0.04}{12}=0.00333$ (monthly interest rate), - $n=30 \times 12=360$ (number of months), $$\begin{gathered} \text { Payment }=250,000 \times \frac{0.00333(1.00333)^{360}}{(1.00333)^{360}-1} \\ \approx 1193.54 \end{gathered}$$ Final Answer: Sarah's monthly payment is $1,193.54.
Scenario: Alex takes out a $\$ 40,000$ student loan at an annual interest rate of $5 \%$, to be repaid over 10 years.
Using the same loan formula, with:
- $P=40,000$,
- $r=\frac{0.05}{12}=0.004167$,
- $n=10 \times 12=120$,
$$
\begin{aligned}
\text { Payment }=40,000 & \times \frac{0.004167(1.004167)^{120}}{(1.004167)^{120}-1} \\
& \approx 424.26
\end{aligned}
$$
Final Answer: Alex's monthly payment is \$424.26.
In the IB Math AI SL course, students are expected to use technology to perform amortization calculations. This primarily involves:
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