Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

Should I lease or buy my first car?

AI SLAI HL Solid Also in: Modelling

Research question

Over four years, is leasing, buying with a loan, or buying a used car with savings the cheapest way to have a car, once depreciation, interest and the opportunity cost of savings are included — and how sensitive is the answer to the interest rate?

Make it yours, then agree it with your teacher.

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Why it makes a good exploration

It is a decision many students face soon after the IB, and the adverts make it deliberately hard to compare. Financial mathematics — loans, depreciation, compound interest — gives a clear, personal answer and a sensitivity analysis that shows when the answer changes.

The mathematics you'll need

Loan amortisation and monthly repayments (TVM solver — explain the formula)
Exponential depreciation fitted to used-car prices
Compound interest and opportunity cost
Break-even analysis and sensitivity to interest rate

Where the data comes from

Collect real quotes for one car model (lease, loan) and 30+ used listings of the same model at different ages to fit depreciation.

ECB Data Portal — exchange rates — Daily euro reference rates for 29 currencies and ECB interest rates.
FRED (Federal Reserve Bank of St. Louis) — 800,000+ economic time series (interest rates, inflation, unemployment) with CSV download.
Desmos graphing calculator — Free graphing and regression (y₁ ~ ax₁ + b) — fit models to your data and show residuals.

Real data for this idea

Primary source: Office for National Statistics (UK): CPI index (series D7BT). The index (2015 = 100) by month, to turn past amounts into today's prices. Licence: Open Government Licence v3.0: free to reuse with an acknowledgement.

If you use FRED (listed above), treat it as link-only: download series for your own analysis, but link to them rather than copying FRED's data into anything you publish. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Define the three options and a fair comparison (same car, same period).
Fit a depreciation model to used-car prices.
Compute total cost of each option, including what the car is worth at the end.
Vary interest rates and mileage to find where the best option changes.
Reflect on risk, flexibility and what the model leaves out (insurance, repairs).

Pitfalls that cost marks

Comparing monthly payments rather than total cost.
Forgetting the car's value at the end of the period.
Quoting rates with no date or source.

Showing personal engagement

Use the car you actually want and real local quotes.
Interview a family member about their car decision.
State the interest rate at which you would change your mind.

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

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
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	Loan amortisation and monthly repayments (TVM solver — explain the formula); Exponential depreciation fitted to used-car prices
AI HL	Good fit	Loan amortisation and monthly repayments (TVM solver — explain the formula); Exponential depreciation fitted to used-car prices

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

Reflection (D)

For this idea, start with the weak points: comparing monthly payments rather than total cost; and forgetting the car's value at the end of the period. Say how each one affects your answer.

Use of mathematics (E)

SL: loan amortisation and monthly repayments (TVM solver — explain the formula); exponential depreciation fitted to used-car prices; compound interest and opportunity cost; break-even analysis and sensitivity to interest rate, each explained in your own words and carried out correctly.

How finance explorations reach the top bands in general: Finance & economics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

HL: fit the depreciation with a logarithmic transformation and a confidence band, and use it to find the age at which buying used is cheapest per year of ownership.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL finance exploration adds.

See a complete IA, marked

Should I lease or buy my first car? (AI SL) explores a question like this one, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL and AI HL. We rate it Solid: it needs some independent work beyond class examples.

What mathematics does this IA use?

Loan amortisation and monthly repayments (TVM solver — explain the formula), exponential depreciation fitted to used-car prices, compound interest and opportunity cost and break-even analysis and sensitivity to interest rate.

Where can I get data for this IA?

Collect real quotes for one car model (lease, loan) and 30+ used listings of the same model at different ages to fit depreciation. Checked sources and tools: Office for National Statistics (UK), ECB Data Portal — exchange rates, FRED (Federal Reserve Bank of St. Louis) and Desmos graphing calculator.

Turn this idea into your IA

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