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

Is the euro–dollar exchange rate a random walk?

AI SLAI HLAA HL Ambitious Also in: Statistics, Probability

Research question

Are daily percentage changes in the EUR/USD reference rate normally distributed and independent, as a simple random-walk model assumes, and how often do extreme days occur compared with the normal model's prediction?

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Why it makes a good exploration

Financial models often assume normally distributed changes, and the 2008 crisis showed why that can be dangerous. Testing the assumption on ECB data is serious statistics about something that affects prices everywhere.

The mathematics you'll need

Percentage change and log returns
Normal distribution and z-scores
Chi-squared goodness-of-fit to a normal
Correlation between consecutive days
Tail probabilities

The statistics, step by step

Worked with every number shown, with what examiners look for and the common mistakes: Chi-squared goodness of fit · Pearson's correlation and regression · Is my data normal?. Then run the same steps on your own data in Analyse my data, or start from the statistics workflow.

Where the data comes from

Download the daily EUR/USD reference rate history from the ECB Data Portal.

ECB Data Portal — exchange rates — Daily euro reference rates for 29 currencies and ECB interest rates.

Real data for this idea

Primary source: ECB Data Portal: Euro foreign exchange reference rates. Choose one currency and download the daily series as CSV. Licence: Free to reuse with the source acknowledged.

Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain the random-walk model.
Compute daily changes and describe them.
Test normality and independence.
Count extreme days vs normal predictions.
Reflect on fat tails and risk management.

Pitfalls that cost marks

Testing prices instead of changes.
Ignoring that ECB rates are daily fixes, not trading prices.
Over-claiming about predictability.

Showing personal engagement

Link to a trip abroad and what the rate cost you.
Compare a calm and a crisis period.
Try (and fail) to beat a random walk with a simple rule.

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	Good fit	Percentage change and log returns; Normal distribution and z-scores
AI SL	Fits — ambitious at SL	Percentage change and log returns; Normal distribution and z-scores
AI HL	Good fit	Percentage change and log returns; Normal distribution and z-scores

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. 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: testing prices instead of changes; and ignoring that ECB rates are daily fixes, not trading prices. Say how each one affects your answer.

Use of mathematics (E)

SL: percentage change and log returns; normal distribution and z-scores; chi-squared goodness-of-fit to a normal; correlation between consecutive days; tail probabilities, 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

Fit a distribution with fatter tails (explain it) and compare its fit with the normal.

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) is a complete finance exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI SL, AI HL and AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Percentage change and log returns, normal distribution and z-scores, chi-squared goodness-of-fit to a normal, correlation between consecutive days and tail probabilities.

Where can I get data for this IA?

Download the daily EUR/USD reference rate history from the ECB Data Portal. Checked sources and tools: ECB Data Portal and ECB Data Portal — exchange rates.

Turn this idea into your IA

The planner opens with “Is the euro–dollar exchange rate a random walk?” filled in: the question, the maths and the data sources.

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