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Maths IA · Modelling · topic 75 of 97

Retirement savings: how long will the money last? · IB Maths IA idea: guide, sample graph and step-by-step plan

interest in, withdrawal out

A fund of 1,000,000 earning 5% a year. Withdraw 50,000 a year and it never runs out. Withdraw 60,000 and it's gone in under 37 years. You're going to find the line between the two, and what inflation does to it.

Difficulty●●○ Medium
KitA spreadsheet, a real or assumed return rate, your GDC
TimeAbout 12 to 15 hours
FitsAA SL3 of 3AA HL3 of 3AI SL3 of 3AI HL3 of 3
  • ✓ you have wondered how much a person needs to retire
  • ✓ you like a formula with a sharp cut-off between lasting forever and running out
  • ✓ you want a model that gets better when you add inflation and risk

The big idea

Each year the fund earns interest and then a withdrawal comes out. Written as a recurrence, the balance is a geometric series, which sums to a formula. Set it to zero and logarithms give the number of years the money lasts. There is a sharp threshold: if the withdrawal is no more than the yearly interest, the balance never falls, and the formula has no solution at all. The setting-up is routine. Real retirements need inflation-linked withdrawals, which change the model into a growing annuity. The marks come from the comparison: how the lifetime changes with the withdrawal rate, which assumption it is most sensitive to, and how a fixed return differs from a risky one.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

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Find the lifetime

Solve for the number of years the fund lasts, for withdrawal rates from 3% to 8% of the starting balance.

→ A log equation and a table. Check it against a year-by-year spreadsheet and explain the rounding of the final year.
The perpetuity threshold

Show when the fund never runs out, and graph years lasting as the withdrawal approaches that threshold.

→ A vertical asymptote, explained from the formula. Say what a retiree should do near it.
Inflation-linked withdrawals

Let the withdrawal grow by a fixed percentage each year to keep its buying power.

→ A second recurrence and a growing-annuity formula. Find how much the safe withdrawal rate drops.
Test a 4% rule of thumb

Compare the rule that a retiree can withdraw 4% of the start balance, adjusted for inflation, with your model for different returns.

→ A rule against a formula. Find the return that makes 4% safe for 30 years and judge how realistic it is.
Withdraw at the start of the year

Switch to withdrawals at the start of each year (as an annuity due) and compare.

→ A shifted formula. Measure how many months the timing is worth.
Random returns

Replace the fixed return by random yearly returns with the same average, and simulate many retirements.

→ A model with chance in it. Estimate the probability of running out before 30 years and say why a bad early year hurts most (HL).

Your Maths IA research question: run the check

which fund? which number?

Write your own question first. Then make sure it ticks every box:

  • Names the fund: starting balance, return rate (real or nominal), withdrawal rule and timing within the year.
  • Names the quantity you will find: years lasting, a safe withdrawal rate or the chance of running out.
  • Says what the formula is tested against: a year-by-year table or published historical returns.
  • Goes beyond a retirement calculator: there is a formula to derive and a threshold to explain.
  • States what you leave out: tax, fees, pensions, changing spending, lifespan uncertainty.
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The model

Each year the fund earns interest, then a withdrawal comes out. Draw how the balance changes for your own fund and mark where it runs out.

Each year: B → (1 + r)B − W (r = return, W = withdrawal)Start: B0 = 1,000,000; r = 5%; W = 6% of B0 each yearyear 0year 10year 20year 300year 40B0if W = rB the bar never shrinksfalls faster and faster:when does it reach 0?years after retiring

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Aim and context~1 page
Why this fund, where the return and withdrawal come from, and what you expect before calculating.
The balance as a sequence~2 pages
Every symbol defined. The yearly step, the geometric series and the closed formula.
How long it lasts~1.5 pages
The log equation, the answer for several withdrawal rates, checked against a table.
The threshold~1.5 pages
When the fund never runs out, and years lasting as a function of the withdrawal rate on one graph.
Inflation-linked withdrawals~3 pages
The growing-annuity model, its lifetime formula and the drop in the safe rate.
Model against published returns~1.5 pages
Your fixed-return answer beside historical or simulated returns, with a number for the gap.
Conclusion~1 page
Back to your aim: how long the money lasts, and what withdrawal you would advise, with numbers.
Evaluation~2 pages, the biggest part!
Fixed returns, tax, fees, spending that changes with age. Say which assumption matters most and test it.
Bibliographynot counted
Where every return, rate and rule came from.

What your Maths IA graph could look like

mark where it hits zero!

A made-up fund to show the form of a strong graph, not the answer: 1,000,000 at 5% a year with three withdrawal sizes. Yours will look different. example only

05101520253035400200400600800100012001400Years since retiring, nBalance B / thousand dollarswithdraw 50,000 a yearwithdraw 60,000 a yearwithdraw 70,000 a yearyour spreadsheet values (made-up)withdrawal equals interest:the balance stays flatILLUSTRATIVE · MADE-UP DATA
  • Balance in thousands of dollars against years, with units on both axes.
  • The flat line is a result: it is where withdrawal equals interest. Derive it.
  • Where each curve meets zero is the answer. Calculate it, then mark it.
  • Plot a few year-by-year spreadsheet rows as points to show your formula is right.
  • Another graph to consider: years lasting against the withdrawal rate, with the threshold marked.

The maths in action

worked through, not sketched!

The graph above was a sketch with made-up numbers. This figure works the idea through from the model itself, so you can see what a finished figure looks like. The source line under it says where every number comes from.

Two panels. Left: balance against years for five withdrawal rates; two rise, one is flat, and two fall to zero after about 37 and 26 years. Right: years lasting against withdrawal rate, rising without limit at 5% for fixed withdrawals and at 3% for inflation-linked ones.
(a) 1,000,000 at 5% a year with fixed withdrawals of 3% to 7% of the start balance: 5% holds flat; 6% lasts 36.7 years, 7% lasts 25.7. (b) Years lasting against the first withdrawal rate, fixed or rising with 2% inflation.Source: Computed from the model (fixed return, withdrawals at the end of each year).

How long does the money last?

Rank the withdrawals on one number: years until the fund is empty. Work it out for your own fund, and explain why the gaps are not equal.

Withdraw 70,00025.7Withdraw 60,00036.7Withdraw 55,00049.15,000 more a year cuts the time sharplyYears the fund lasts, 1,000,000 at 5% a yearILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Read and predictFind out how a drawdown fund works. Before any calculation, guess how long 1,000,000 lasts when you withdraw 6% a year.
  2. Choose the fundPick a balance, a return and a withdrawal rule you can justify: a real pension plan, a published return or a typical case where you live.
  3. Write the yearly stepDefine every symbol and write the balance after one year. Say whether withdrawals come at the start or the end of the year.
  4. Trial run (1 lesson)Build a 10-year spreadsheet and compare it with your formula. Fix any mismatch now: timing, rate, rounding.
  5. Sum the seriesSpot the geometric series and derive the closed form. Solve for the number of years with logarithms.
  6. Find the thresholdShow when the fund never runs out and graph years lasting against the withdrawal rate.
  7. Add inflationLet the withdrawal grow each year, derive the new lifetime formula and compare it with the fixed case.
  8. Go further, with a reasonTest a 4% rule of thumb or add random returns. HL: this is where the rigour for criterion E comes from.
  9. Reflect as you goAfter every result, write one line on what it means for a retiree and what it assumes. These lines become your evaluation.
  10. Cut, check, then sendCheck every graph has labelled axes with units, every symbol is defined once and the conclusion answers your aim. Then send your teacher one draft.

The maths you'll use

  • Recurrence and geometric series – Write and sum the series to get .
  • Logarithms – Set and solve for ; show the equation has no solution when .
  • Infinite geometric series – Explain the perpetuity threshold: the present value of withdrawals forever is , so the fund lasts forever only if .
  • Growing annuity – Replace by and derive the lifetime .
  • Random returns (HL) – Simulate yearly returns from a normal distribution with the same mean and estimate the chance of running out within 30 years.

Key equations

Syllabus links

  • 1.3Geometric sequences and series
  • 1.4Compound interest and depreciation
  • 1.7Rational exponents and laws of logarithms
  • 1.8Infinite geometric series
  • 2.9Exponential and logarithmic functions
  • 4.9Normal distribution

Read up first

Data and sources to start from

Use a published long-run return and inflation rate for the base case, and either historical yearly returns or a clearly stated simulation for the risk extension.

References to cite

  1. Bengen, W. P. (1994). Determining withdrawal rates using historical data. Journal of Financial Planning, 7(4), 171-180.
  2. Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
  3. Cooley, P. L., Hubbard, C. M., & Walz, D. T. (1998). Retirement savings: Choosing a withdrawal rate that is sustainable. AAII Journal, 20(2), 16-21.

Cite what you actually use, in your own format, and say where each number came from.

Traps to avoid in this Maths IA

real or nominal? say which
  • Mixing a nominal return with real withdrawals. Say whether your rate has inflation taken out, and be consistent.
  • Forgetting the threshold. If the withdrawal is at most the interest, the log formula fails; say what that means.
  • Rounding the lifetime without saying how the last year works: a part-year withdrawal is not a full one.
  • Taking a historical rule of thumb as a law. It describes one set of past returns, not the future.
  • Using the average return in a risky model and calling it safe. Two funds with the same average can end very differently.
  • Quoting a retirement calculator's output without deriving the formula behind it.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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