Loan Repayments and Amortization in Math AI
Understanding Amortization
- Amortization is a financial concept that refers to the process of gradually paying off a debt over time through regular payments.
- In the context of loans, it involves spreading out the repayment of the principal (the original amount borrowed) and the interest over a set period.
Amortization is crucial in many real-world financial situations, including mortgages, car loans, and student loans.
The amortization process typically follows these steps:
- Calculate the total payment amount (covering both principal and interest)
- Apply a portion of the payment to the interest owed
- Apply the remainder to reduce the principal balance
- Recalculate the interest for the next period based on the new, lower principal balance
As the loan progresses, the proportion of each payment going towards the principal increases, while the interest portion decreases.
ExampleMortgage Payment Calculation
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.
Student Loan Repayment
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.
Using Technology for Amortization Calculations
In the IB Math AI SL course, students are expected to use technology to perform amortization calculations. This primarily involves:
- Graphic Display Calculators (GDCs)
- Spreadsheet software (e.g., Microsoft Excel, Google Sheets)
Graphic Display Calculators
Most modern GDCs come with built-in financial packages that can handle amortization calculations. Common functions include:
- TVM (Time Value of Money) Solver
- Amortization schedules
- Interest rate conversions
Using a TI-84 Plus calculator:
- Press [APPS] and select "Finance"
- Choose "TVM Solver"
- Input the loan details: N (number of payments) I% (annual interest rate) PV (present value / loan amount) PMT (payment amount, if known) FV (future value, usually 0 for loans)
- Solve for the unknown variable (often PMT)
Spreadsheets
Spreadsheet software offers more flexibility and visual representation of amortization schedules.
