IA idea · Simulation & Monte Carlo methods

Will the pension pot last? Monte Carlo savings with random returns

AI SLAI HL Ambitious Also in: Finance, Statistics

Research question

If yearly investment returns are random rather than fixed, what is the probability that a saver reaches a target pot, or that a pot lasts 30 years of withdrawals, and how misleading is a fixed-rate calculation?

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

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Why it makes a good exploration

Financial maths in class uses one interest rate; real returns vary. Simulating thousands of possible futures shows the risk that a single number hides.

The mathematics you'll need

Geometric series for a fixed-rate annuity
Modelling yearly returns with a normal distribution fitted to historical data
Simulation of many paths
Probability of success and percentiles
Effect of the order of returns (sequence risk)

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 historical yearly returns of a market index or interest rates; FRED publishes many financial time series. State exactly which series you used.

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.

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

Do the fixed-rate calculation.
Fit a distribution to historical returns and check it.
Simulate many savings paths.
Compare success probabilities with the fixed-rate answer.
Reflect on fat tails, inflation and fees.

Pitfalls that cost marks

Fitting a normal distribution without checking it.
Mixing nominal and real (inflation-adjusted) returns.
Too few simulated paths for the tail probabilities.

Showing personal engagement

Use the savings plan of someone you know (anonymised).
Find the withdrawal rate that succeeds 95% of the time.
Compare a cautious and an adventurous mix.

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	Fits — ambitious at SL	Geometric series for a fixed-rate annuity; Modelling yearly returns with a normal distribution fitted to historical data
AI HL	Good fit	Geometric series for a fixed-rate annuity; Modelling yearly returns with a normal distribution fitted to historical data

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

Personal engagement (C)

Design the simulation yourself: choose the rules, test them on a small case you can check by hand, and change one rule at a time to answer a question you care about.

Reflection (D)

Compare simulation with exact theory or real data, say how many runs you used and how much the answer varies between batches, and question the random-number assumptions. For this idea, start with: fitting a normal distribution without checking it — say how it affects your answer.

Use of mathematics (E)

SL: A probability model described precisely, simulated correctly, with the simulated answer compared with an exact calculation for at least one simple case and the results summarised with appropriate statistics.

HL: An estimate of the simulation's error (standard error, or a confidence interval for the estimate), a distribution fitted to the results and tested, or an exact result proved for the general case that the simulation confirms.

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

Resample real historical years instead of a fitted distribution and compare the answers.

Extending it for HL

Give a confidence interval for each simulated estimate and show how it narrows as the number of runs grows, or prove a general result that the simulation confirms.

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 simulation 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).