Bills, Notes and Bonds: What a Yield Does to a Price

Buy one Treasury security at par on a day in 2026, revalue it on another, and watch the price move. The numbers are Treasury's own daily curves, nominal and inflation-protected.

Worked example — Bills, Notes and Bonds: What a Yield Does to a Price

The 2026 long end, measured	4 September
Yield move	+38 bp
Real yield (TIPS)	+33 bp
Breakeven inflation	+5 bp
Price per 100 face	94.28
Change since purchase	−5.72%
Modified duration	15.70
Basis points per 1% of price	6

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Bills, Notes and Bonds: What a Yield Does to a Price

A bond is a promise to pay fixed amounts on fixed dates. Its price and its yield are one fact stated two ways: the yield is the single discount rate that makes those promised payments add up to that price. Fix the payments and the two can only move in opposite directions.

What each symbol means

P: price per 100 of face
c: the annual coupon, paid in two halves
y: the yield per year
n: the number of payments still to come
D: modified duration, the percentage the price falls when the yield rises by one percentage point

Assumes: That the payments arrive. A US Treasury carries no default premium in its yield, which is what makes it the clean instrument for this lesson. Prices here are struck on a coupon date, so there is no accrued interest, and each security is bought at par, so its coupon equals the yield on its purchase date.
Breaks when: Duration is a straight line laid against a curve, so it overstates a loss and understates a gain, and the error grows with the size of the move: 0.02 percentage points on the 2-year this year, 0.25 on the 30-year. That bend is convexity, and the chart draws both lines so you can watch them part. Duration also assumes the whole curve shifts together, which it did not in 2026, when the front end moved more than twice as far as the long end.

The long end rose least in 2026 and lost the most 🖖

Over the eight months to 4 September the 2-year yield rose 90 basis points and the 30-year rose 38. Buy each at par on 2 January and revalue them on the September curve: the 2-year is worth 98.29 and the 30-year 94.28. The long bond lost 3.35 times as much money from 0.42 times the yield move. Duration is the whole reason, and the tool prints it — 1.92 years against 15.70. Move the security selector down the list and watch the loss grow while the yield move shrinks.

Almost none of it was inflation 🖖

Treasury publishes a second curve, for inflation-protected securities, and the gap between the two is the inflation rate the market is pricing. Of the 38 basis points the 30-year added this year, 33 came from the real yield, which went from 2.63% to 2.96% and touched 3.06% on 17 August. Five came from expected inflation. Fear of inflation is the usual explanation for a high long yield, and here it is doing almost none of the work.

Thirty-year inflation is priced below ten-year inflation, and has been all year 🖖

On 4 September the breakeven was 2.35% at ten years and 2.28% at thirty. That is the wrong way round for anyone expecting inflation to compound away over decades, and it is not one day's quirk: it held on 170 of the 171 sessions in the file. Select the 10-year, read the breakeven card, then select the 30-year.

A price of 94 is not a loss unless you sell 🖖

The 30-year at 94.28 still pays 100 at maturity, and every coupon in between. The loss belongs to a seller. That is why one bank can carry the same bond at 94 and another at 100, depending on whether it is held to maturity, and it is the distinction that decided Silicon Valley Bank in 2023. What the price card shows is what somebody else would pay you today.

Every maturity on the curve now costs more than the debt already pays 🖖

The average interest rate across all interest-bearing federal debt was 3.447% on 31 July. The cheapest point on the 4 September curve is the one-month constant maturity at 3.79%, and the thirty-year is 5.24%. As old debt matures and is reissued the average climbs toward the curve, which happens whether or not a single yield ever moves again.

Problem solved in full

Which loses more: 90 basis points on a two-year, or 38 on a thirty-year? 7 steps
Both were bought at par on 2 January 2026. Over the eight months to 4 September the 2-year's yield rose 90 basis points and the 30-year's rose 38, less than half as far. Work out which lost more money, and by how much, before reading the price cards.
1. Duration is a weighted average of when a bond's money arrives, each payment weighted by its share of the price. A two-year note pays almost all of it inside two years. A thirty-year bond has coupons landing three decades out, and those distant payments are the ones a discount rate has most power over.
  
  $D = \frac{\sum _{t} t \frac{C _{t}}{( 1 + i ) ^{t}}}{\sum _{t} \frac{C _{t}}{( 1 + i ) ^{t}}}$
2. For a bond bought at par there is a closed form, so there is no need to sum sixty terms by hand. With i the yield per half-year and n the number of half-years, Macaulay duration is ((1 + i) / i) × (1 − (1 + i)⁻ⁿ), measured in half-years.
  
  $at par : D = \frac{1 + i}{i} (1 - (1 + i)^{- n}) half-years$
3. The two-year: i = 3.47 / 200 = 0.01735 and n = 4. That gives 3.8988 half-years, so 1.9494 years.
  
  $i = \frac{3 . 47}{200} = 0.01735, n = 4 : D = 3.8988 \to 1.9494 years$
4. The thirty-year: i = 4.86 / 200 = 0.02430 and n = 60. That gives 32.1709 half-years, so 16.0854 years. Eight times the two-year's figure, from fifteen times the maturity.
  
  $i = \frac{4 . 86}{200} = 0.02430, n = 60 : D = 32.1709 \to 16.0854 years$
5. Modified duration divides each by (1 + i): 1.9162 and 15.7038. That is the number the question needs, because a price change is roughly −D times the yield change.
  
  $D_{mod} = \frac{D}{1 + i} : 1.9162 and 15.7038$
6. The two-year: −1.9162 × 0.90% = −1.725%. The thirty-year: −15.7038 × 0.38% = −5.967%.
  
  $\frac{Δ P}{P} \approx - D_{mod} Δ y : - 1.9162 \times 0 . 90% = - 1 . 725%, - 15.7038 \times 0 . 38% = - 5 . 967%$
7. Now compare the exact prices: −1.71% and −5.72%. Duration overstated both, by 0.02 percentage points on the two-year and 0.25 on the thirty-year.
  
  $2-year 30-year Δ y + 0 . 90% + 0 . 38% duration says - 1 . 725% - 5 . 967% actual - 1 . 71% - 5 . 72% error 0.02 0.25$
Answer

The thirty-year loses 5.72% against the two-year's 1.71%, a ratio of 3.35 from 0.42 times the yield move. Duration alone predicted 3.46, and the gap is convexity: the price-yield relationship is a curve, duration is the straight line tangent to it at the purchase point, and the longer the bond the sooner that line leaves the curve. Open the price-and-yield panel to see the two drawn together. The error is one-sided — the curve bends upward everywhere, so duration always overstates a loss and always understates a gain.

$\frac{5 . 72}{1 . 71} = 3.35 from \frac{0 . 38}{0 . 90} = 0.42$

References (2)

The two curves this tool is built from, and how they are constructed — which is why its figures are closes rather than intraday quotes: U.S. Department of the Treasury, "Interest Rate Statistics" — Daily Treasury Par Yield Curve Rates and Daily Treasury PAR Real Yield Curve Rates.
The long-end intervention the tool's opening section describes, in Treasury's own words: U.S. Department of the Treasury, "Treasury Announces Increased Sizes of Nominal Long-End Liquidity Support Buybacks Beginning September 9", press release, 19 August 2026.

By the 42Math team · Reviewed against published sources. Corrections welcome.