Room to be wrong
Buying a home ties a large part of your life to one decision. The price matters. So do the cash left after buying, the bills you can absorb, and what you can learn before the decision becomes expensive to undo.
Start with a €400k apartment. Follow its owner through uncertain expenses, then go back to the decision to buy.
A 10% fall can cost you 40%.
Buy a €400,000 apartment with €100,000 of your own money and a €300,000 mortgage. Your equity is the home’s value minus what you owe. Watch what happens when the price falls 10%, then rises 10%.
Debt amplifies the change in your equity. A €40k fall in the home’s price takes 40% of your €100k. Then a 10% recovery acts on the smaller home value: €360k becomes €396k, leaving €96k of equity. The price path and the financing both matter.
You can finish ahead and run out on the way.
Now separate equity from cash. You have €100k after moving and purchase costs. Put €94k into the same €400k home and keep €6k available. Your mortgage is €306k. After ordinary living costs and loan payments, you expect €500 a month left over.
Extra expenses and occasional large bills interrupt that plan. In this invented example, they average €410 a month. On average, cash grows by €90 a month. But your bills arrive in a particular order.
Keeping cash has a cost. So does needing it.
Split the same €100k differently. A smaller down payment leaves more cash and a larger mortgage. In this model, every extra €10k borrowed costs €33 a month at 4% interest. Move money between the home and the bank account.
This is the trade-off behind leaving a cushion after buying. Money in the home reduces borrowing; money in the bank can pay a bill. Neither role replaces the other.
An inspection can buy the choice to walk away.
A cash buffer helps after you own the home. Before buying, you can sometimes pay to find out more while you can still walk away.
A house costs €30k less than an otherwise suitable alternative. But it needs work. The discount is only a saving if the work costs less than €30k. Assume €2k of inspections and specialist checks reveals the full cost, while you can still choose either home.
First, the work turns out to cost €42k. Buying the cheaper house would leave you €12k worse off. Compare buying immediately with checking first and choosing the alternative.
A less predictable house can make information worth more.
Keep the €30k discount. First, repair costs sit in a narrow range: €24k–€28k. Then widen that range to €2k–€50k. Both average €26k, so the expected saving before checking stays at €4k.
The checks still cost €2k. Widen the unknowns and watch which purchases you can avoid. Each dot is one possible repair bill for this house.
Knowing is useful only if you can still act.
Keep the same uncertain repair bills. Make the checks more expensive, then try finding out after you have already committed to buying. The report can be equally accurate while being much less valuable.
- Average saving if you buy unchecked
- €4k
- Losses you can avoid by walking away
- Cost of finding out
- Average after checking
The worst possible result still counts. Here it is .
The model also assumes the checks reveal every repair cost. A valuation or a tidy interior cannot establish that. For an apartment, the building’s future work and shared finances can matter too. An inspection and a valuation answer different questions; the ability to withdraw depends on the agreement and local rules.
Before buying, useful information can help you refuse a loss. After buying, available cash can help you live through one. Both have a price. The practical question is how the purchase changes what you can afford to do next.
Assumptions and calculations
The home, the mortgage, and the monthly budget
All prices, rates, probabilities, and inspection fees are invented teaching assumptions, expressed in euros. They are not estimates of a housing market or a mortgage offer. The first example holds debt at €300k while a €400k home falls 10% and then rises 10%. It excludes transaction costs, mortgage payments, taxes, and sale costs.
The budget starts with €100k available after moving and purchase costs. A €6k buffer leaves €94k for the down payment and a €306k mortgage. The €500 monthly surplus is after ordinary spending and baseline loan payments. Each month adds an independent extra cost drawn uniformly from €0–€400. There is also an independent 1.5% chance of an extra €8k–€20k bill, drawn uniformly. This could stand for repairs or another household setback; it is not an estimated frequency of either.
Average extra costs are €200 + 0.015 × €14,000 = €410 per month. The average surplus is €90, so expected cash after five years is €6,000 + 60 × €90 = €11,400. The model generates 10,000 paths of 60 months from a fixed seed. A funding gap occurs the first time a monthly balance falls below zero. Later balances continue arithmetically to expose gaps hidden by an ending balance; no overdraft, rescue loan, or recovery is assumed to be available.
Keeping more cash increases the mortgage by the same amount. We charge 4% annual interest on that extra debt, with no principal repayment on the increment and no interest earned on the cash. Every future pays that additional interest. Mortgage eligibility, changes in the quoted rate, taxes, inflation, refinancing, sale proceeds, and wider household choices are outside this example. A funding gap alone does not tell us whether a real owner would default.
How this connects to ergodicity and fragility
Peters and Gell-Mann study why growth along a single wealth history can differ from averages across possible outcomes. Their multiplicative-growth models motivate looking at the process through time. Our monthly cash model illustrates a related but distinct problem: a positive expected ending balance can hide an earlier funding gap. It does not calculate an ergodic growth rate or prove a general ranking of home-buying strategies.
The first example combines percentage changes with debt. If home value is V and debt is D, equity is V − D. A price change ΔV changes equity by ΔV, so the percentage change in equity is ΔV ÷ (V − D). The mortgage magnifies both gains and losses on the initial equity.
The inspection example below has a convex payoff. It benefits from our mean-preserving increase in spread only while the ability to reject a costly purchase remains intact. This is an illustration of Taleb and Douady’s response-based account of fragility and antifragility. These examples do not need fat-tailed distributions.
What the inspection can and cannot establish
The house is €30k cheaper than a suitable alternative. All other relevant costs and benefits are held equal. Repairs are the only uncertainty. The first two examples use €42k and €6k of work: net savings of −€12k and +€24k. The chart then compares repair costs uniform on [€24k, €28k] with [€2k, €50k]. Both average €26k.
In the model, the checks reveal the full repair bill perfectly, and the alternative stays available at the same price. Before commitment, the buyer can choose that alternative at no penalty. Buying this house makes sense within the model only when the discount exceeds the repair bill. Real inspections may miss defects, costs can change, other buyers may compete, and the right to withdraw depends on the contract and jurisdiction. The CFPB sources describe US practice; our euro examples assume no particular country’s lending or contract rules.
Under Already committed, the full purchase result is yours regardless of what the report says. Checking subtracts its fee without avoiding any purchase loss. This isolates the value of the ability to act, rather than the informational value of a report for planning repairs.
The payoff, the dots, and the averages
Let X be the €30k discount minus repair costs, and c the cost of checking. Buying unchecked gives X. Checking while free to choose gives max(X, 0) − c. Checking after commitment gives X − c. The reference alternative is zero in each comparison.
For X uniform on [−20, 28], in thousands of euros, the expected positive part is 28² ÷ (2 × 48) ≈ 8.17. Subtract the €2k fee to get €6.17k. Equivalently: €4k average saving + €4.17k avoided losses − €2k checks. This is the expected value of one purchase decision, not a long-run wealth forecast. Choosing which home to investigate next is a separate problem studied by Weitzman.
The cash chart shows a fixed sample of 1,200 futures (600 on narrow screens). The payoff chart shows 1,000 repair bills (500 on narrow screens), each under two decisions. The sorting view shows all 10,000 bills. All summaries use the full simulation or the stated analytical distribution. Jitter is always on: position offsets separate dots without changing the calculations. Exact amounts are available by selecting a dot. Within the sorting groups, position carries no numerical value.
Dot identities persist through 650-millisecond, interruptible transitions; reduced motion disables them. Main plot marks are vectors, so zooming keeps their edges sharp. The 10,000-dot sorting view uses the display’s full pixel resolution.