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.
ExampleCreating a basic amortization schedule in Excel:
- Set up columns for: Payment Number, Payment Amount, Principal Paid, Interest Paid, Remaining Balance
- Use formulas to calculate each column:
- Interest Paid = Remaining Balance * (Annual Interest Rate / 12)
- Principal Paid = Payment Amount - Interest Paid
- New Remaining Balance = Previous Remaining Balance - Principal Paid
- Copy formulas down to see the entire loan lifecycle
Experiment with different loan scenarios by changing variables like interest rate or loan term to see how they affect the total interest paid and monthly payments.
End-of-Period Payments
In IB Math AI SL examinations, it's important to note that payments are assumed to be made at the end of each period. This is known as "ordinary annuity" in financial terms.
$$ \text{Present Value} = \text{Payment} \times \frac{1 - (1 + r)^{-n}}{r} $$
Where:
- $r$ is the interest rate per period
- $n$ is the number of periods
While understanding this formula enhances comprehension, students are not expected to memorize or directly use it in exams.
Link to Exponential Models
The concept of loan repayments and amortization is closely related to exponential models, covered in SL 2.5. This connection is evident in the way interest compounds over time.
Consider a loan balance $B$ after $t$ years with an annual interest rate $r$:
$$ B = P(1 + r)^t $$
Where $P$ is the principal amount.
HintThis exponential growth of the loan balance (if left unpaid) is counteracted by regular payments in the amortization process.
Common MistakeStudents often confuse simple and compound interest. Remember that most loans use compound interest, which grows exponentially, not linearly.
Real-World Applications
Understanding loan repayments and amortization is crucial for various real-world financial decisions:
- Mortgages: Comparing different loan terms and interest rates to find the most affordable option.
- Student Loans: Calculating the long-term impact of education debt and planning repayment strategies.
- Car Loans: Evaluating the true cost of vehicle financing over time.
- Credit Card Debt: Understanding the high cost of carrying balances and the importance of timely payments.
- Retirement Planning: Calculating how much to save regularly to reach a target retirement fund.
Comparing two mortgage options:
- 30-year fixed at 3.5% interest
- 15-year fixed at 2.8% interest
Using a GDC or spreadsheet, students can calculate and compare:
- Monthly payments
- Total interest paid over the loan term
- Equity buildup over time
