Interest rates as the price of everything else

A number moves by a quarter of a percentage point. Six weeks later, on the other side of the world, a warehouse that was going to be built is not built, a company that was going to be bought is not bought, and a couple who were going to move house decide to stay another two years.

None of them talked to each other. All of them were doing the same arithmetic.

The arithmetic that does all the work

Suppose something pays 100 a year, forever, and never changes. What is it worth?

It depends entirely on what else money can earn. If the going rate is 5%, then 2,000 put anywhere else would also produce 100 a year, so the thing is worth about 2,000. If the going rate is 10%, then only 1,000 is needed to produce 100 a year elsewhere, so the thing is worth about 1,000.

The income did not change. Nothing about the asset changed. The rate doubled and the value halved.

That is the whole mechanism, and everything else in this lesson is a wrinkle on it.

Predict

Rates rise sharply. Four things are repriced. Which falls in value by the most?

Choose one answer

A loan that repays everything in twelve monthsA loan that repays a fixed amount in thirty yearsCash in an accountA shop that makes a steady profit every year and always has

Lock in answer

From the borrower's side

The same number seen from underneath is the cost of having something before you have paid for it. Here it is on a long loan, which is where most households meet it.

Everything below is held fixed except the rate: the same building, the same deposit, the same twenty-five years.

Property loan

≈ Hypothetical model, not a forecast

A long-dated secured loan: the payment, the interest bill, and what paying extra each month changes.

It assumes

• The rate is fixed for the whole twenty-five years, which very few real long loans are.
• Every payment is made in full and on time, with no missed month anywhere.
• There are no fees, no purchase taxes and no cost of arranging the loan.
• The building itself never changes in value, and the model says nothing about whether it would.

It ignores

• Purchase taxes, legal costs, survey fees and lender fees
• Insurance, maintenance, and anything else the property costs to hold
• Any change in the value of the property itself
• Rate resets, which is how most long loans actually behave

Find the rate at which the total interest bill first exceeds the amount borrowed — the point where the lender is paid more for the use of the money than the money itself.

Annual interest rate5%

Amount borrowed

240,000

Deposit paid up front

60,000

Loan as a share of the price

0.8

Scheduled monthly payment

1,403

Total handed over, scheduled payments only

420,905

Total handed over, paying extra

420,905

Total interest, scheduled payments only

180,905

Total interest, paying extra

180,905

Months taken when paying extra

300

Months removed from the term

0

Interest avoided

0

• Balance outstanding
• Interest paid so far

◷ Not there yet — keep moving the controls.

The number that crosses over sits a little above six percent. Below it the building costs more than the borrowing; above it the borrowing costs more than the building. Nothing about the building was involved in that.

Why the price of time is also the price of risk

A quoted rate is not one thing. It is a stack of separate charges bundled into one number, and taking the stack apart explains most of what looks arbitrary about lending.

The pure price of waiting. What a lender would need even if repayment were certain, because they would rather have it now.

Expected inflation. If prices are expected to rise, money repaid later buys less, and the lender wants compensating for that before anything else.

Credit risk. The chance this particular borrower does not repay, multiplied by how much would be lost.

Liquidity and term. Being unable to get the money back for a long time is itself a cost, and it grows with the length of the loan.

Two borrowers quoted very different rates are usually not being treated differently on the first two components at all. The gap is almost entirely the third and fourth, which is why a rate quoted to a person carries information about how they are seen rather than about the money.

Check

A business is deciding whether to spend 500,000 on equipment that will save 40,000 a year. The rate at which it can borrow rises from 6% to 9%. What has changed?

Choose one answer

The project's return is unchanged, but the bar it has to clear went upThe equipment became more expensiveNothing, if the business pays cash insteadThe saving of 40,000 is now worth less in real terms

Check answer

Who sets it

Partly a central bank, and partly not, and the split matters.

A central bank sets a very short rate directly — the price of money lent overnight between institutions. Everything longer is set in markets, by what lenders and borrowers will agree, and long rates embed expectations about what short rates will do for years. So the lever is real and it is short and blunt, and the number that actually decides whether a warehouse gets built is a long one that the lever influences without controlling.

›Why a low rate makes long-dated things go strangeOptional

When the rate is very low, the arithmetic at the top of this lesson starts producing large numbers.

A stream of 100 a year at 5% is worth 2,000. At 2% it is worth 5,000. At 1% it is 10,000. The valuation is not misbehaving; it is doing exactly what it always did, and the function gets steep as the rate approaches zero.

Two consequences follow. Small changes in a low rate produce enormous changes in the value of long-dated assets, so those assets get much more volatile. And assets whose payoff is furthest away — a business expected to earn nothing for a decade and a great deal afterwards — go from marginal to spectacular on a rate move alone, and back again.

Worth remembering

• An interest rate is the price of moving money through time, and it sits inside the value of anything that pays in the future.
• Doubling the rate roughly halves the value of a fixed income stream without anything about the asset changing.
• The further away an asset's money is, the more its value moves when rates move.
• A quoted rate bundles the price of waiting, expected inflation, credit risk and the cost of being locked in.
• A rate change moves the threshold a project must clear, so projects near the boundary stop happening without anyone deciding to stop them.