# Amortization and Annuities

> IB Mathematics Applications and Interpretation SL · IB Math AI SL
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u1-amortization-and-annuities/

We cover ordinary annuities, annuities due, amortization formulas, TVM solver workflows, and common exam calculation scenarios for regular payment financial products.

**Prerequisites:** [Compound interest calculations](https://www.owlsprep.com/study/ib-math-ai-sl-u1-compound-interest/); [Sum of finite geometric series](https://www.owlsprep.com/study/ib-math-ai-sl-u1-geometric-series/)

## Learning objectives

- Distinguish between ordinary annuities and annuities due
- Apply the amortization formula to calculate regular loan repayments
- Compute total interest paid over the term of an amortizing loan
- Use the TVM financial solver to find unknown annuity variables

## Core Annuity Definitions

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.

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

*Notation:* End-of-period payments

**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

**Check your understanding**

Test your understanding of basic annuity classification

1. 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

   *Why:* Payments made at the start of the billing period are classified as annuities due

## Present Value and Amortization

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: $PMT = 20000 \times \frac{0.005}{1-(1.005)^{-60}}$
4. Evaluate to get the final monthly repayment of \$386.66

> **tip**
>
> For a fully amortized loan, the future value input in your TVM solver will always be set to 0, as the remaining balance is reduced to zero at the end of the term.

## Future Value of Annuities

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: $FV = 100 \times \frac{(1.004)^{120} - 1}{0.004}$
4. Final accumulated value after 10 years is \$15,390.19

> **info**
>
> Note that the total amount you personally deposited is only \$12,000, the remaining \$3,390.19 is compound interest earned over the term.

## Annuities Due and TVM Solver Workflows

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 | $PMT \times \frac{1-(1+r)^{-n}}{r}$ | Multiply result by $(1+r)$ |
| FV | $PMT \times \frac{(1+r)^n -1}{r}$ | 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

**Exam command terms**

IB exam questions use specific command terms for annuity problems

- **Find the regular repayment** — You are expected to use the amortization formula or TVM solver, show full inputs *(2022 Paper 1 Q7)*

- **Calculate the total interest paid** — Subtract the initial loan principal from total sum of all repayments

## Common pitfalls

- **Wrong:** Using the annual nominal interest rate directly instead of dividing by number of compounding periods
  - Why it fails: Fails to match periodic payment frequency to the correct per-period interest rate, leading to wildly incorrect results
  - Correct: Always calculate r as nominal annual rate divided by number of payments per year before substituting into formulas
- **Wrong:** Confusing PV and FV inputs for loan calculations
  - Why it fails: Loans use PV (the initial sum borrowed) not FV, and setting PV to 0 will return a mathematically invalid repayment value
  - Correct: Set FV = 0 for all fully amortized loan calculations, and enter the initial loan amount as PV
- **Wrong:** Forgetting to adjust for annuities due
  - Why it fails: Payments made at the start of the period earn one extra period of interest, so unadjusted ordinary annuity results will be off
  - Correct: Multiply all ordinary annuity PV and FV results by (1 + r) to get the correct annuity due value
- **Wrong:** Rounding intermediate values during multi-step calculations
  - Why it fails: Final repayment values can be off by more than \$10, leading to lost accuracy marks in the exam
  - Correct: Store full unrounded values in your GDC for all steps, only round the final answer to 2 decimal places for currency
- **Wrong:** Ignoring partial final payments for non-integer term scenarios
  - Why it fails: Examiners will deduct marks if you give equal full payments that overpay the loan balance
  - Correct: Calculate the remaining balance after n-1 full payments to find the exact partial final payment value

## Cheatsheet

| Calculation | Ordinary Annuity (end of period) | Annuity Due (start of period) |
| --- | --- | --- |
| Present Value | $PV = PMT \times \frac{1-(1+r)^{-n}}{r}$ | $PV = PMT \times \frac{1-(1+r)^{-n}}{r} \times (1+r)$ |
| Future Value | $FV = PMT \times \frac{(1+r)^{n}-1}{r}$ | $FV = PMT \times \frac{(1+r)^{n}-1}{r} \times (1+r)$ |
| Amortization PMT | $PMT = PV \times \frac{r}{1-(1+r)^{-n}}$ | $PMT = PV \times \frac{r}{1-(1+r)^{-n}} \times \frac{1}{1+r}$ |

## 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.

---

From [OwlsPrep](https://www.owlsprep.com) — free study guides for A-Level, IB, AP and IGCSE, written against the official syllabus. Canonical page: https://www.owlsprep.com/study/ib-math-ai-sl-u1-amortization-and-annuities/