Amortization and Annuities

IB Mathematics Applications and Interpretation SL· 18 min read

1. Core Annuity Definitions★★☆☆☆⏱ 4 min

All annuity calculations are derived directly from the sum of a finite geometric series, where each payment earns compound interest for the remaining term of the product. The two most common annuity types tested in IB AI SL are ordinary annuities and annuities due.

📘 Definition

Ordinary Annuity

End-of-period payments

Payments are made after the interest for the period has been applied. This is the default assumption for most standard loan and savings products.

📐 Worked Example

Identify which of the following scenarios describes an ordinary annuity: A) Rent paid on the first day of each month, B) Mortgage repayment paid on the last day of each month

1
An ordinary annuity requires payments at the end of each period
2
Scenario A (rent) uses start-of-month payments, so it is an annuity due
3
Scenario B (mortgage) uses end-of-month payments, so it is an ordinary annuity

✓ Quick check

Test your understanding of basic annuity classification

Which of these is an annuity due?
- Weekly grocery spending
- Annual salary paid at the end of the month
- Phone bill paid on the first day of the billing cycle
Reveal answer
Phone bill paid on the first day of the billing cycle —
Payments made at the start of the billing period are classified as annuities due

2. Present Value and Amortization★★★☆☆⏱ 5 min

Amortization calculations use the present value of an ordinary annuity to find the equal regular payment required to pay off an initial lump sum loan completely over a fixed term. The total amount repaid will always be larger than the initial loan, as you are charged compound interest on the remaining balance each period.

PMT = PV \times \frac{r}{1-(1+r)^{-n}}

📐 Worked Example

Calculate the monthly repayment for a $20,000 car loan, with 6% annual nominal interest compounded monthly, paid off over 5 years

1
First calculate the periodic monthly interest rate: r = 0.06 / 12 = 0.005
2
Calculate total number of periods: n = 5 * 12 = 60 months
3
Substitute values into the amortization formula: $P M T = 20000 \times \frac{0.005}{1 - ( 1.005 ) ^{- 60}}$
4
Evaluate to get the final monthly repayment of $386.66

3. Future Value of Annuities★★★☆☆⏱ 4 min

The future value of an annuity calculates the total accumulated sum after making regular equal payments into an interest-earning account over a fixed term. This is most commonly used for savings scenarios like retirement funds or education savings plans.

FV = PMT \times \frac{(1+r)^n - 1}{r}

📐 Worked Example

Calculate the total future value after 10 years if you deposit $100 at the end of each month into an account earning 4.8% annual interest compounded monthly

1
Calculate monthly interest rate: r = 0.048 / 12 = 0.004
2
Total number of periods: n = 10 * 12 = 120 months
3
Substitute into future value formula: $F V = 100 \times \frac{( 1.004 ) ^{120} - 1}{0.004}$
4
Final accumulated value after 10 years is $15,390.19

4. Annuities Due and TVM Solver Workflows★★★★☆⏱ 5 min

Annuities due shift every payment one period earlier, so each payment earns one extra period of compound interest compared to an ordinary annuity. This means you can adjust all ordinary annuity formulas by multiplying by (1 + r) to get the equivalent annuity due result.

Variable	Ordinary Annuity	Annuity Due Adjustment
PV	$P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}$	Multiply result by $(1 + r)$
FV	$P M T \times \frac{( 1 + r ) ^{n} - 1}{r}$	Multiply result by $(1 + r)$

📐 Worked Example

Calculate the future value of the earlier $100 monthly savings scenario if payments are made at the start of each month instead of the end

1
Take the ordinary annuity FV result of $15,390.19
2
Multiply by (1 + r) = 1.004 to adjust for annuity due
3
Final adjusted future value is $15,451.75

5. Common Pitfalls

Wrong move:

Using the annual nominal interest rate directly instead of dividing by number of compounding periods

Why:

Fails to match periodic payment frequency to the correct per-period interest rate, leading to wildly incorrect results

Correct move:

Always calculate r as nominal annual rate divided by number of payments per year before substituting into formulas

Wrong move:

Confusing PV and FV inputs for loan calculations

Why:

Loans use PV (the initial sum borrowed) not FV, and setting PV to 0 will return a mathematically invalid repayment value

Correct move:

Set FV = 0 for all fully amortized loan calculations, and enter the initial loan amount as PV

Wrong move:

Forgetting to adjust for annuities due

Why:

Payments made at the start of the period earn one extra period of interest, so unadjusted ordinary annuity results will be off

Correct move:

Multiply all ordinary annuity PV and FV results by (1 + r) to get the correct annuity due value

Wrong move:

Rounding intermediate values during multi-step calculations

Why:

Final repayment values can be off by more than $10, leading to lost accuracy marks in the exam

Correct move:

Store full unrounded values in your GDC for all steps, only round the final answer to 2 decimal places for currency

Wrong move:

Ignoring partial final payments for non-integer term scenarios

Why:

Examiners will deduct marks if you give equal full payments that overpay the loan balance

Correct move:

Calculate the remaining balance after n-1 full payments to find the exact partial final payment value

6. Quick Reference Cheatsheet

Calculation	Ordinary Annuity (end of period)	Annuity Due (start of period)
Present Value	$P V = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r}$	$P V = P M T \times \frac{1 - ( 1 + r ) ^{- n}}{r} \times (1 + r)$
Future Value	$F V = P M T \times \frac{( 1 + r ) ^{n} - 1}{r}$	$F V = P M T \times \frac{( 1 + r ) ^{n} - 1}{r} \times (1 + r)$
Amortization PMT	$P M T = P V \times \frac{r}{1 - ( 1 + r ) ^{- n}}$	$P M T = P V \times \frac{r}{1 - ( 1 + r ) ^{- n}} \times \frac{1}{1 + r}$

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2023 · Paper 2
Monthly annuity repayment calculation
2022 · Paper 1
Total interest for car loan amortization
2021 · Paper 2
Retirement annuity future value

What's Next

Mastering amortization and annuities gives you a critical foundation for the financial mathematics questions that make up 15-20% of your IB Math AI SL Paper 2 marks. You will now be able to approach common real-world scenarios including mortgage calculations, student loan repayments, and regular retirement savings planning with confidence. The next logical step is to practice applying these formulas to unfamiliar multi-part exam questions that combine annuities with percentage change and inflation adjustments. You can also build on this knowledge to explore related financial topics including currency exchange, bond valuation, and depreciation of assets, which are all regularly tested in the IB AI SL assessment. Make sure to complete the full practice question set for this sub-topic to lock in your calculator workflow before your mock exams.