Maths IA · Modelling · topic 80 of 97
Rent or buy: when does buying a home win? · IB Maths IA idea: guide, sample graph and step-by-step plan
same scale: today's moneyBuying costs a deposit and fees up front, and then more each year than renting. Yet in the model below it overtakes renting after about 8 years. You're going to find the break-even year, and what moves it.
- ✓ your family has argued about whether to rent or buy
- ✓ you like comparing two plans on one fair scale
- ✓ you want to test which input a big decision really depends on
The big idea
Money now is worth more than money later, so you cannot compare rent and a mortgage by adding up dollars. Discount every payment back to today at a rate you choose, and each stream of cash flows becomes one number, its present value. Rent is a growing stream; buying is a deposit, a mortgage, upkeep, and a home you sell at the end. The discounting sums are geometric series, so they have closed forms. The break-even year is the first year buying's total cost falls below renting's. The result depends on assumptions such as price growth and the discount rate. The marks come from testing them: which input moves the break-even most, and how fragile is your conclusion?
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.
Swipe for more twists →
Write the present value of renting for N years and of buying and selling after N years, with every symbol defined.
→ Two geometric series and a loan balance formula. Check both against a year-by-year spreadsheet.Set the two totals equal and solve for N, with a solver or by interpolation between years.
→ An equation you cannot solve by algebra. Explain what the answer means for a family that moves after N years.Change the mortgage rate, house-price growth, rent growth and discount rate in turn, and record the new break-even year.
→ A sensitivity table. Rank the inputs and say which assumption your conclusion rests on.Estimate how many years the break-even moves per percentage point of each input, using a small change or a derivative.
→ A rate of change of a result, not of a function you wrote down. Say where the slope is steep and why.Replace yearly steps by continuous flows and integrals, with discount factor e^(−δt).
→ Sums become integrals. HL work: compare the integral model with the yearly one and measure the difference.Take real rent, price and mortgage figures for one city and recompute the break-even.
→ Your model meets data. Any gap between your assumptions and the real numbers becomes part of your evaluation.Your Maths IA research question: run the check
which city? which number?Write your own question first. Then make sure it ticks every box:
- Names the choice: the home price, rent, deposit, mortgage rate and term, real or assumed with a reason.
- Names the quantity you will find: the break-even year, or the input the conclusion depends on most.
- Says what the discount rate stands for, and why you chose that value.
- Says what the model is tested against: a year-by-year table or real rent and price data.
- States what you leave out: taxes, moving costs, repairs that vary, the value of living in your own home.
The model
Two timelines of cash flows, one for renting and one for buying. Draw them for your own case and mark every payment you will discount.
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.
Why this choice, where the numbers come from, and what you expect before calculating.
Why a dollar later is worth less, the discount factor, and present value of one payment and of a growing stream.
Rent growing each year as a geometric series, summed to a closed form and checked against a table.
Deposit and fees, mortgage payments, upkeep and the sale. The loan balance as a formula.
The break-even year, then one input changed at a time, with one graph and one table.
Back to your aim: when buying wins, and how fragile that is, with numbers.
Taxes, repairs, moving costs, a rate that is not constant. Say which assumption matters most and test it.
Where every rent, price and rate came from.
What your Maths IA graph could look like
mark the break-even!A made-up case to show the form of a strong graph, not the answer: a 500,000 home, 20% deposit, mortgage at 4.5% over 25 years. Yours will look different. example only
- Both totals in today's money on one set of axes, with units on both axes.
- The break-even year is where the curves cross. Calculate it, then mark it.
- Buying starts higher because of the deposit and fees: say why it falls below renting.
- Plot spreadsheet rows as points to show the closed forms are right.
- Another graph to consider: the break-even year against one input, such as the mortgage rate.
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.
How much does the mortgage rate matter?
Re-solve for the break-even year each time one input changes, then rank the cases. Work it out for your own numbers, and explain why the rise is so sharp.
Your game plan
Tick steps off as you go. We'll remember on this device.
0 of 10 done
- Read and predictFind out what present value means and how a rent-or-buy comparison works. Before any calculation, guess the break-even year for your case.
- Choose the casePick a home price, rent, deposit and mortgage you can justify: real listings, a published rate or a typical case where you live.
- Choose the discount rateDecide what a dollar tied up in the home could earn elsewhere, give a reason, and keep it fixed for the first model.
- Trial run (1 lesson)Build a 10-year spreadsheet for both options. Fix any mismatch now: timing of payments, the loan balance, selling costs.
- Derive the closed formsSum the rent and upkeep series, and write the loan balance after N years. Check each against the table.
- Find the break-evenSolve for the year when buying's total cost drops below renting's. Say how you handle a part year.
- Test the inputsChange one input at a time. Record the break-even and rank the inputs by their effect.
- Go further, with a reasonEstimate slopes of the break-even or build a continuous model. HL: this is where the rigour for criterion E comes from.
- Reflect as you goAfter every result, write one line on what it means for a family and what it assumes. These lines become your evaluation.
- 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
- Present value – Discount each cash flow by and add them, so two streams of payments can be compared in today's money.
- Geometric series – Sum the growing rent, , and the growing upkeep to closed forms.
- Loan balance formula – Find the mortgage balance after years, , to get the money recovered when selling.
- Solving for the break-even – Set the two present values equal and solve for with a solver, or interpolate between years, and check it in a table.
- Sensitivity analysis – Change one input at a time and record the break-even; estimate the slope by a small change.
- Integrals (HL) – Model rent as a continuous flow and find its present value .
Key equations
Syllabus links
- 1.3Geometric sequences and series
- 1.4Compound interest and depreciation
- 1.8Infinite geometric series
- 2.9Exponential and logarithmic functions
- 2.10Solving equations graphically and analytically
- 5.11Definite integrals and areas between curves
- 1.3Geometric sequences and series
- 1.4Compound interest and depreciation
- 1.7Loans, amortisation and annuities
- 1.8Solving systems and polynomial equations with technology
- 5.5Integration as anti-differentiation; areas
Read up first
Data and sources to start from
Take real rents, home prices and a mortgage rate for one city or country from published statistics, and say how each was chosen.
- FRED: 30-year fixed rate mortgage average in the United States
- FRED: S&P CoreLogic Case-Shiller U.S. National Home Price Index
- Housing & Development Board (Singapore)
References to cite
- Brealey, R. A., Myers, S. C., & Allen, F. (2020). Principles of corporate finance (13th ed.). McGraw-Hill Education.
- Kellison, S. G. (2009). The theory of interest (3rd ed.). McGraw-Hill/Irwin.
- Himmelberg, C., Mayer, C., & Sinai, T. (2005). Assessing high house prices: Bubbles, fundamentals and misperceptions. Journal of Economic Perspectives, 19(4), 67-92.
Cite what you actually use, in your own format, and say where each number came from.
Traps to avoid in this Maths IA
discount before you compare- Adding up dollars from different years without discounting. The comparison is meaningless until they are all in today's money.
- Leaving out the deposit's opportunity cost. The discount rate is what captures it, so choose it with care and say so.
- Counting the deposit as spent forever. If you sell, you get back the home's value minus the loan left.
- Choosing one set of inputs and calling the break-even a fact. It is a result of your assumptions.
- Mixing a nominal discount rate with real rent growth. Keep every rate on the same basis.
- Forgetting selling and buying costs, which move the break-even by years.
More Maths IA ideas like this
Home loans: where does each payment go?●●○
Retirement savings: how long will the money last?●●○
Loans and savings: when does a plan break even?●○○
Flat rate or effective rate: what does a car loan really cost?●●○
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.