The Discount Rate: Why a Dollar in 2035 Is Worth Less Today

Someone offers you a choice. A hundred dollars today, or a hundred dollars in 2035. You take today's without thinking, and you're right. But ask the next question, the one a tycoon asks: how much would the 2035 offer have to be before you'd take it instead? A hundred and ten? A hundred and fifty? Two hundred?

Whatever number you land on, you've just done the most important calculation in finance, and you've done it with a rate in your head. The rate is the price of waiting. Every stock, every bond, every building, every business, every pension on earth is priced by exactly that calculation, run by a few million people at once, using a rate that gets printed in public every day.

This course is about that rate: where it comes from, what moves it, and what it does to everything you own when it moves. The question this lesson answers: what's a promise of money in the future actually worth right now, and who sets the price?

Pisa, 1202. A merchant's son named Leonardo, thirty or so, has come home from years in the trading ports of North Africa and the eastern Mediterranean, where he learned arithmetic from Arab teachers. He writes a book. It's called Liber Abaci, the book of calculation, and it's remembered today for one thing, a sequence of numbers about rabbits.

The rabbits are one page. The rest of the book is trade: how to split profits among partners, how to convert currencies, how to price a bill of exchange, how to compare two ways of being paid. And in one problem, the most sophisticated in the book, he does something no one had written down before.

The problem is titled, in the translation the historian William Goetzmann quotes, On a Soldier Receiving Three Hundred Bezants for his Fief. A soldier has a pension from the king: three hundred bezants a year, paid seventy five at a time, every quarter. The king changes the terms. He'll pay the same three hundred, but all at once, at the end of the year. Same money. Is the soldier worse off?

Leonardo says yes, and shows how much. The soldier could have invested each quarterly payment as it arrived, at the going rate, which the problem sets at two bezants per hundred per month. Every payment that comes later than it used to is money that couldn't be earning in the meantime. He works the delay into a number and states the soldier's true loss.

A payment moved later is worth less, and the difference has a price, and the price can be calculated exactly. That's the whole of present value, written in a Pisan ledger eight hundred years ago. Goetzmann's paper argues Leonardo was the first person in history to set it down.

Seven centuries later an American economist, Irving Fisher, wrote the formula the way it's taught now. By then every banker in the world was using it without knowing whose it was.

Ask the tycoon's question of the king. Who made the rule? The king, who changed the terms. Who gained? The king: he kept the soldier's money for most of a year and earned on it. Who paid? The soldier, by exactly the amount Leonardo computed. The rule was made by the one who held the money longer. It usually is.

Here's the formula, in words first, then in numbers. It's simple enough to do on a napkin and it prices everything.

Step one is the promise. Someone owes you a dollar on a future date. A bond pays a coupon next year. A tenant pays rent every month. A company earns a profit every quarter. A pension pays out at sixty five. Every one of these is a list of promises, each with a date.

Step two is the rate. Ask: what could I earn, safely, on a dollar between now and that date? For a dollar in the United States the answer is printed every trading day: the yield on a Treasury bond of that length.

Call it the discount rate, which means the rate you use to shrink a future promise down to its value today. It's called that because the shrinking is a discount, the way a bill paid early gets one.

Step three is the discount itself. Take the promise and divide it by one plus the rate. That's what the promise is worth if it's due a year from now. If it's due in two years, divide again. Nine years, nine divisions.

The present value, the number you get at the end, is the amount you'd need today, invested at the rate, to have the promised dollar on the promised date.

Run it. The ten-year Treasury yields four point nine five percent this week. A dollar due in 2035, nine years away, divided by one point oh four nine five nine times, comes to about sixty five cents. That's the price of a 2035 dollar today. Not an opinion. Arithmetic on a public number.

Step four is the sum. A stock is a long list of future profits. A building is a long list of future rents. Discount every one of them by its own wait and add them up, and the total is what the thing is worth. Every price on the board is a sum of future promises, each one shrunk by the rate, and the rate is the only number in the sum that everyone shares.

Which means one more thing, and it's the thing this course turns on. Change the rate and every sum changes at once, without a single promise being broken. Lower the rate and every future dollar gets bigger today; every price rises. Raise it and every future dollar shrinks; every price falls. Nothing about the businesses changed. The price of waiting did.

The rate has a chart, and it's the most important one in this course.

The ten-year yield since 1962. It climbed from four percent to nearly sixteen in September 1981, the top of the last long inflation, then fell for forty years, to a floor of about half a percent in August 2020. Then it turned. By the autumn of 2023 it was back near five, and this week it printed four point nine five.

Now do what Leonardo did, to the same promise, at each of those rates.

One dollar, due in 2035. At the 2020 rate it was worth ninety five cents today: the future was nearly free, and a promise nine years out was almost as good as cash. At this week's rate the same dollar is worth sixty five cents. At the 1981 rate it would have been twenty seven cents.

Read that table as a history of every asset price you've ever seen. From 1981 to 2020 the rate fell and fell, and every future dollar got more valuable today, year after year, without any business having to earn more.

That's forty years of rising prices for stocks, houses, bonds, and anything else that was a list of future promises.

Then in 2022 the rate went from one and a half to near five in eighteen months, and the price of a 2035 dollar fell by about a third. So did the prices of the things made of them.

The incentive map. Who sets the rate? For the short end, a committee, as the money course taught. For the ten-year, the market: every lender and borrower in the world, arguing in public. Who gains when it falls? Everyone who already holds a future promise. Who pays when it rises? Everyone who bought a future promise at the old price, and everyone who has to borrow at the new one.

Here's the reveal. The rate isn't one price among many. It's the price of the future itself, and when it moves, every other price on earth moves with it, whether or not the thing behind the price changed at all. The forty-year fall from 1981 to 2020 wasn't a golden age of business. It was a golden age of discounting.

A digital tycoon owns future promises on both sides of his ledger, and this is the lesson that tells him what they're worth. Three reads.

First, your business is a present value. The number a buyer will pay for it's the sum of the profits they expect, each one discounted at the rate on the day they buy. When the rate doubles, your company is worth less on paper even if every customer stayed.

The office prices its own business the same way and re-runs the sum every quarter, so the number on the board is never a wish.

Second, know which of your holdings are far away. A promise due next year barely moves when the rate moves; a promise due in twenty barely survives it. The lesson after this one is about that difference, and it's the reason the same rate move hits two stocks so differently. For now, write next to each holding when its money actually arrives.

Third, watch the ten-year, not the headlines. The committee moves the overnight rate and the news reports it. The market moves the ten-year, and the ten-year is what discounts your life.

When the two disagree, as they do this week with the ten-year more than a point above the overnight rate, the market is telling you what it thinks the future costs, and it's the market's number that prices the board.

Our paper fund reads it by its rules: real prices, a written reason on every move, no leverage, and a score measured against Bitcoin alone, an asset with no coupon to discount, which is exactly why the desk keeps it as the ruler.

• Leonardo of Pisa (Fibonacci, c. 1170 to 1240) published Liber Abaci in 1202 in Pisa and revised it in 1228. Goetzmann argues the book contains the first known present value analysis. NBER Working Paper 10352, March 2004.
• The problem "On a Soldier Receiving Three Hundred Bezants for his Fief": an annuity of 300 bezants a year paid in quarterly installments of 75, changed by the king to a single year-end payment of 300, with the soldier able to earn 2 bezants per hundred per month; Leonardo computes the soldier's loss from the delay. Goetzmann, quoting Sigler's translation.
• "The modern present value formula was developed by economist Irving Fisher in 1930." Goetzmann.
• 10-year Treasury constant maturity yield (DGS10): 4.06 percent on January 2, 1962; 15.84 percent on September 30, 1981; 0.52 percent on August 4, 2020; 1.52 percent on December 31, 2021; 4.98 percent on October 19, 2023; 4.95 percent on September 10, 2026. FRED.
• Present value of one dollar due in nine years, at each rate: 0.52 percent, 95.4 cents; 2 percent, 83.7 cents; 4.95 percent, 64.7 cents; 8 percent, 50.0 cents; 15.84 percent, 26.6 cents. Arithmetic: 1 divided by (1 + rate) to the ninth power.
• The fall in the value of a nine-year dollar from the December 31, 2021 rate (1.52 percent: 87.3 cents) to the October 19, 2023 rate (4.98 percent: 64.6 cents) is 26 percent. Arithmetic on FRED.