IA idea · Numerical methods & error analysis

What interest rate are you really paying? Solving for the rate by iteration

AI SLAI HLAA SL Solid Also in: Finance

Research question

Given a real loan's amount, monthly payment and term, what interest rate does it imply, and how quickly do bisection and Newton–Raphson find it when there is no formula?

Adapt it: change the place, the data or the comparison until the question is yours.

Free: the A–E checklist an examiner uses, by email ↓

Why it makes a good exploration

The annuity formula can't be rearranged for the interest rate, so financial calculators iterate. Doing it yourself explains what the TVM solver does and lets you check a real advertised rate.

The mathematics you'll need

Annuity formula from a geometric series
Why the rate can't be isolated algebraically
Bisection with an error bound
Newton–Raphson with a derivative of the annuity function
Comparing with the GDC's financial solver

Course labels show where a technique sits; using maths from outside your course is fine if you explain it clearly and say it is new to you.

Where the data comes from

Use a real advertised loan or finance deal (car, phone, student) with its terms; record the date and source.

Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Cite every source in a footnote where you use it and in your bibliography. Check the licence of any dataset you download.

A possible outline

Derive the annuity formula.
Show why the rate can't be isolated.
Solve with bisection and track the error.
Solve with Newton–Raphson and compare speed.
Check the advertised rate and reflect on fees and how rates are quoted.

Pitfalls that cost marks

Confusing monthly and annual rates.
Using a solver without explaining it.
Ignoring arrangement fees that change the true rate.

Showing personal engagement

Check a deal your family was offered.
Find a deal where the advertised rate looks misleading.
Explain the result to whoever took the loan.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Good fit	Annuity formula from a geometric series; Why the rate can't be isolated algebraically
AA HL	Not a natural fit	The mathematics is mainly from the AI course; at AA HL the exploration would need an AA-level approach (calculus, proof or probability theory) to reach the top of Criterion E.
AI SL	Good fit	Annuity formula from a geometric series; Why the rate can't be isolated algebraically
AI HL	Good fit	Annuity formula from a geometric series; Why the rate can't be isolated algebraically

Level: Solid. Needs some independent work beyond class examples. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Personal engagement (C)

Choose the function or equation yourself and predict how each method will behave before you run it. Hunting for the cases that break a method is engagement an examiner can see.

Reflection (D)

Reflect on error: how it changes with step size or iterations, why a method converges slowly or not at all, and how you know your 'exact' comparison value is correct. For this idea, start with: confusing monthly and annual rates — say how it affects your answer.

Use of mathematics (E)

SL: The trapezoidal rule or a simple iteration applied correctly, errors tabulated against step size and explained, with any method outside the syllabus explained step by step.

HL: Convergence analysed rather than observed: an error bound derived with calculus or a series, an order of convergence measured and justified, or Euler's method studied against an exact solution.

Criteria A and B (presentation and communication) work the same way for every idea: see the guides to Criterion A and Criterion B.

Taking it further

Compare APR conventions in two countries, or find the rate for a loan with irregular payments.

Extending it for HL

Derive an error bound with a Maclaurin series or calculus, then show your numerical results follow it.

See a complete IA, marked

Our annotated exemplar Should I lease or buy my first car? (AI SL) asks a different question, but shows how a complete numerical methods exploration is structured and marked, with an examiner's comment on every criterion. Free excerpts and the full marking table are on its page.

Before you start: the checklist an examiner uses

Every check for Criteria A–E in a 4-page PDF, the mistakes that cost the most marks and a self-assessment grid. We'll email it with a short IA tip every few days, timed to your deadline if you give it. Free — no account, no payment.

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Turn this idea into your IA

Plan this IAOpens the planner with this question, the maths and data sources filled in. Criteria checklistCheck your plan or draft against the A–E descriptors and self-grade before you hand it in. AI feedback on a draftCriterion-by-criterion feedback with evidence from your own text. Coaching, not rewriting. Examiner reviewPete, an IB examiner, reads your draft and talks it through with you (€100).