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IB Mathematics AI SL Amortization and annuities using technology Study Notes - New Syllabus

IB Mathematics AI SL Amortization and annuities using technology Study Notes

LEARNING OBJECTIVE

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Key Concepts: 

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MAI HL and SL Notes – All topics

Graphic Display Calculator TI-84 Plus Codes

♦The TVM (Time-Value-of-Money) Solver in TI-84 Plus is a useful tool for calculating compound interest problems. 

Example

You invest $5000 in a savings account with an annual interest rate of 5%, compounded monthly, and want to know the value of your investment after 10 years.

▶️Answer/Explanation

Solution:

 Step 1:
Press the APPS button on the calculator, then select TVM Solver.

 Step 2:
Enter the following values:

  • N = 120 → (12 months × 10 years)

  • I/Y = 5 → (Annual interest rate as a whole percentage)

  • PV = -5000 → (Initial investment; entered as a negative value because it’s a cash outflow)

  • PMT = 0 → (No additional payments)

  • FV = ? → (We want to find this)

  • P/Y = 12, C/Y = 12 → (Monthly compounding)

Note: Ensure that P/Y and C/Y are both set to 12, since interest is compounded monthly.

 Step 3:
Move the cursor to the FV field, press ALPHA, then ENTER (SOLVE).
The calculator will compute:

FV = $\$8,235.05$

♦Code:

  • APPS → TVM Solver

  • Enter:

    • N = 120

    • I/Y = 5

    • PV = -5000

    • PMT = 0

    • FV = ?

    • P/Y = 12, C/Y = 12

  • Move to FV, press ALPHA → ENTER

  • Result: FV = $8,235.05

Amortisation and Annuities

♦Amortisation

Amortisation is the process of gradually paying off a debt with regular payments, where each payment is divided between the principal amount and the interest.

The formula for the monthly payment \(P\) for an amortised loan of principal \(A\), with interest rate \(r\) and term \(n\) is: \(P=\frac{rA}{1-(1+r)^{-n}}\)
The total amount paid over the term of the loan is nP, and the total interest paid is $nP − A$

 Formula for Monthly Payment \(P\)

\(P = \frac{rA}{1 – (1 + r)^{-n}}\)

Where:
\(A\) = Loan principal
\(r\) = Monthly interest rate (annual rate ÷ 12)
\(n\) = Total number of payments

Total Amount Paid: \(nP\)
Total Interest Paid: \(nP – A\)

Example

Amortisation Example:

 Loan: \$10,000 for 5 years at 6% annual interest.

▶️Answer/Explanation

Solution:

\(P = \frac{(0.06/12) \times 10,000}{1 – (1 + 0.06/12)^{-60}} \approx \$193.33\)
Total Paid: \(60 \times 193.33 = \$11,599.80\)
Total Interest: \(11,599.80 – 10,000 = \$1,599.80\)

♦Annuities

An annuity is a series of equal periodic payments made at the end of each period.

 Present Value (\(PV\)) of an Ordinary Annuity:

\(PV = \frac{P}{r} \left[ 1 – \frac{1}{(1 + r)^n} \right]\)

Future Value (\(FV\)) of an Ordinary Annuity:

\(FV = P \cdot \frac{(1 + r)^n – 1}{r}\)

Example

Annuity Example:
Deposit: $\$500$ monthly at 4% annual interest for 5 years.

▶️Answer/Explanation

Solution:

\(FV = 500 \cdot \frac{(1 + 0.04/12)^{60} – 1}{0.04/12} \approx \$33,163.62\)
Balance After 5 Years: $\$33,163.62$

Note: The TI-84 Plus calculator can be used to solve for payment, present value, or future value of an annuity using the TVM Solver function.

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