Duration: Why the Future Moves First on Rate Days

Lesson 3.1 gave you the formula: a future dollar is worth less today, and the rate sets how much less. Now here's the puzzle that formula creates.

In 2022 the rate went up for everyone at once. The same rate, the same year, the same country. And the results were nowhere near the same. A two-year government note barely noticed. A broad basket of American stocks lost about a fifth. A basket of technology stocks lost a third. And one government bond, from a country that has never missed a payment, lost three quarters of its value.

Same rate. Same year. Nobody defaulted. The question this lesson answers: why does the same rate move barely touch one holding and cut another in half?

Vienna, late June 2020. The Republic of Austria sells a bond that repays in the year 2120. A hundred years. The coupon is zero point eight five percent a year. The price is ninety eight, meaning ninety eight cents on the dollar, for a yield of zero point eight eight.

Investors line up for it. In a world where the ten-year American bond yields half a percent, a century of Austria at nearly one looks like income.

Six months later, in December, Austria comes back and sells more of the same bond, and the price is a hundred and thirty five. Rates have fallen further since June, and a promise of zero point eight five a year for a century is now worth a premium to face. Anyone who bought in June is up by more than a third on a government bond in half a year.

Nothing about Austria changed. The price of a century changed.

Then 2022. Rates rise across the world, fast. Austria's coupon doesn't move: it's zero point eight five, in writing, until 2120. But the promise is a hundred years long, and the rate that discounts it has gone from under one percent to over three.

By the autumn of 2023 the bond trades near thirty four. From the December 2020 tap, that's a fall of about seventy five percent. By this summer it's under thirty.

Every coupon was paid on time. Austria's credit was never in question. A bond from a country that never missed a payment lost three quarters of its price, because the only thing that changed was the distance to the money and the price of waiting for it.

Ask the tycoon's question. Who made the rule? The buyers in 2020, who paid a premium for a century of income at less than one percent. Who gained? Austria, which locked in a hundred years of borrowing at under one percent. Who paid? Every holder who needs the money before 2120, which is every holder. The rule was made by the borrower, and the borrower held the pen for a century.

The tool for this has a name, and once you have it you'll see it everywhere.

Duration is the distance to your money. Take any holding, list every dated promise it makes, and compute the average distance in years to those promises, weighted by how much money arrives at each date. That average is the duration.

A two-year note that pays a coupon and then repays has a duration a bit under two. A century bond with a tiny coupon has almost all its money at the far end, so its duration is enormous: sixty seven years at issue, by the arithmetic.

Here's why the number matters. Duration is also a lever. A rate change of one percentage point moves the price by about the duration, in percent, the other way.

Rates up one point on a two-year note: price down about two percent. Rates up one point on a sixty-seven-year bond: price down, roughly, two thirds. That's not a coincidence of the Austrian bond. It's the formula from lesson 3.1, run on every promise and summed.

Think about why. A promise due next year is divided by one plus the rate once. Change the rate and the answer barely moves. A promise due in sixty years is divided sixty times over. Change the rate and the answer collapses or explodes. The farther away the money, the more of its present value is discount rather than promise, so the more of it the rate controls.

Now the step that makes this a lesson for tycoons rather than bond traders. Stocks have duration too. A company's price is the sum of its discounted future profits. A company earning big profits now has most of its value near: short duration. A company that earns nothing yet but is expected to earn a fortune in 2035 has all its value far: long duration. It's a century bond with a logo.

And there's a third kind of holding, the one this shelf keeps coming back to: a thing with no dated promises at all. Gold. Land held for its own sake. A bitcoin. Its duration isn't long or short; the formula doesn't apply, because there's no coupon to discount.

That doesn't make it safe from the rate. It makes it priced by something else, scarcity and demand, and the office keeps that difference written next to the position.

That's why the same year hit a mature index for a fifth and the technology index for a third. The technology index was longer. Its money was farther away. The rate did more of the work, so the rate did more of the damage.

The sort is the skill. Line up everything you own by how far away its cash is. That order is the order the next rate move will hit it in, and the sizes of the hits are roughly the distances. You've just done in a sentence what the bond desk does with a spreadsheet.

One caution the desk writes in the margin. Duration tells you the size of the hit for a given rate move. It doesn't tell you the rate move. A long book is a bet that the rate falls or holds; a short book is a bet it rises or holds.

Neither is wrong. What's wrong is holding a long book without knowing it's long, which is how the century-bond buyers of 2020 found out what sixty seven years feels like.

The rate that did it. The ten-year Treasury yield from the 2020 floor through the 2022 repricing: from half a percent in the summer of 2020, to one and a half at the end of 2021, to nearly four a year later, to nearly five in the autumn of 2023. The two-year moved even more, from under one percent to over four in a single year.

Now the scoreboard for that calendar year, sorted the way the mechanism says to sort it.

The two-year note, two years of distance, lost a few percent as its yield went from under one to over four. The ten-year, about eight years of distance, lost something like a sixth.

The S&P 500, a basket of businesses earning now and later, lost nineteen point four percent. The Nasdaq, a basket earning mostly later, lost thirty three point one. And the century bond, sixty seven years of distance, lost three quarters from its peak.

Read down the column. The losses are in order of distance, and they're roughly proportional to it. Nobody in that table missed a payment. Nobody's business broke. The rate rose, and every price fell by about as much as its money was far away.

The incentive map. Who set the rate? The committee, for the short end; the market, for the long. Who gained from the rise? Anyone holding cash and short paper, who could now lend it at four instead of nothing. Who paid? Everyone who had bought distance when distance was nearly free: the century-bond buyers of 2020, the buyers of profitless growth in 2021, the pension funds that matched a hundred-year promise with a hundred-year bond.

Here's the reveal, and it's the one that separates people who understand the board from people who watch it. In 2022 the market didn't decide technology companies were worth less. It decided the future was worth less, and technology companies were where the future was kept. The rate didn't judge the businesses. It repriced the distance.

A digital tycoon holds long-duration things by nature: his own business, whose profits are mostly ahead of it, and the digital future, whose whole thesis is that the money arrives later. The office reads the lesson three ways.

First, know your own duration. Write next to every holding when its cash actually arrives. A business paying you now is short. A business you're building for a sale in five years is long. A coin with no cash flow at all is a special case, priced on scarcity rather than promise, which is why the desk keeps it as a ruler and not as a bond.

Second, size by distance. The office's rules cap what can be lost on any day. Duration tells you which holdings can lose the most on a rate day. A long book on a day the ten-year jumps half a point is a bad day by arithmetic, not by luck. Size the far end so that day doesn't end the game.

Third, buy distance when it's expensive to hold, not when it's free. When the future was nearly free everyone bought it, and the year the rate rose was the bill. Today's rate discounts a 2035 dollar to sixty five cents. That's a much cheaper future than the one sold in 2020. The office doesn't forecast the rate. It notes what distance costs today, and it's less than it was.

Our paper fund reads it by its rules: a written reason on every move, no leverage, and a book it can hold through a rate day, because a machine that has to sell on the day the future reprices is the machine from last week's game film.

• Austria's 0.85 percent bond maturing in 2120 was issued June 24, 2020 at a price of 98.010 for a yield of 0.880 percent, with a Macaulay duration of 67.4 years at issue. A tap on December 11, 2020 priced at 135.233. By early October 2023 it traded near 34, about 75 percent below the December 2020 level. On July 28, 2026 it closed at 29.510 with a yield of about 3.3 percent. Summitward.
• The June 2020 buyer's gain to the December tap: 135.233 divided by 98.010 is 1.38, about 38 percent. Arithmetic on the source.
• Treasury yields, December 31, 2021 to December 30, 2022: 2-year 0.73 to 4.41 percent; 10-year 1.52 to 3.88 percent; 30-year 1.90 to 3.97 percent. FRED.
• 10-year yield: 0.52 percent on August 4, 2020; 4.98 percent on October 19, 2023. FRED.
• S&P 500: 4,766.18 on December 31, 2021; 3,839.50 on December 30, 2022; −19.4 percent. Nasdaq Composite: 15,644.97 to 10,466.48; −33.1 percent. FRED.
• Duration approximation: price change in percent is about minus the modified duration times the change in yield in percentage points. For the 10-year Treasury in 2022 (duration roughly 8 to 9 years, yield up 2.36 points) that implies a price fall in the high teens before the coupon; the two-year (duration under 2, yield up 3.68 points) a fall of several percent. Standard fixed-income arithmetic; the exact figures depend on the specific bond.
• Present value of a dollar due in nine years at 4.95 percent: 64.7 cents. Lesson 3.1.