Why must a bond's price move when interest rates move?

You met the seesaw in Markets Foundations as a story: new bonds pay more, so old bonds get cheaper. True, and enough to pass a quiz. This lesson replaces the story with the arithmetic, because once you can compute the new price, the seesaw stops being a fact you remember and becomes a fact you can derive.

The forced move, in one sentence

The coupon is fixed by contract, so when the market's required return changes, the only variable left free to move is the price.

A bond is a claim on a fixed set of future payments. If buyers now demand 7% to part with their money, and your bond's contract can only ever pay $50 a year on $1,000, then the only way your bond can deliver 7% to a new buyer is if the new buyer pays less than $1,000 for it. Nothing else in the contract can flex. The price is the adjustment mechanism, and it is not optional.

Pricing a bond properly: discount every payment

A bond's fair price is the present value of all its future cash flows, discounted at the market's current required yield.

Present value of one payment:

PV = payment ÷ (1 + y)ⁿ

where y is the yield and n is the number of periods away. Sum this over every payment, and you have the price. That's the whole model — no more.

Worked example: rates rise

Take a 3-year bond, $1,000 face, 5% annual coupon. At issue, when the market required 5%, it was worth exactly par: $1,000.

Now the market's required yield rises to 7%. Its cash flows are unchanged: $50, $50, $1,050. Discount each at 7%:

• Year 1: 50 ÷ 1.07 = $46.73
• Year 2: 50 ÷ 1.07² = 50 ÷ 1.1449 = $43.67
• Year 3: 1,050 ÷ 1.07³ = 1,050 ÷ 1.225043 = $857.11

Price = 46.73 + 43.67 + 857.11 = $947.51, or 94.75 in quote terms.

The bond lost $52.49, about 5.2% of its value. Nobody sold in a panic. No news came out about the issuer. The discount rate changed, and the arithmetic did the rest.

Worked example: rates fall

Same bond, but the required yield drops to 3%:

• Year 1: 50 ÷ 1.03 = $48.54
• Year 2: 50 ÷ 1.03² = $47.13
• Year 3: 1,050 ÷ 1.03³ = 1,050 ÷ 1.092727 = $960.90

Price = $1,056.57, or 105.66. The bond gained $56.57.

Note the asymmetry: a 2-point fall in yield gained $56.57, while a 2-point rise lost only $52.49. Gains are slightly larger than losses for equal-sized yield moves. That curvature is called convexity, and it falls straight out of dividing by (1+y)ⁿ — you have just derived it from arithmetic rather than being told it.

Why longer bonds swing harder

Look at where the damage happened in the first example. At a 5% yield the year-3 payment was worth 1,050 ÷ 1.05³ = $907.03; at 7% it is worth $857.11. That one payment accounts for about $50 of the $52.49 lost. Distant payments are punished hardest by a higher discount rate, because the (1+y)ⁿ in the denominator compounds.

Push that to 30 years and it becomes dramatic. A single $1,000 payment 30 years away:

• At 3%: 1,000 ÷ 1.03³⁰ = 1,000 ÷ 2.4273 = $411.99
• At 5%: 1,000 ÷ 1.05³⁰ = 1,000 ÷ 4.3219 = $231.38

A 2-point rate change nearly halves the value of that payment. This is why a long government bond — free of any default worry — can lose a third of its price in a rate-rising year. Safe from default is not safe from rates. Practitioners compress this sensitivity into a single number called duration, which a later course builds properly.

The rule, now earned

• Yields up → prices down. Always, mechanically.
• Yields down → prices up. Always, mechanically.
• Longer maturity → bigger swing, because distant cash flows are more sensitive to discounting.

This is not a market tendency or a historical pattern. It is division. It cannot fail to be true, and nothing here says anything about which way yields will go — only what the price must do if they do.

In the data

The seesaw shows up as two lines moving in opposite directions. The first chart is the US 10-year government yield, in per cent, so 4.70 means 4.70%. The second is a fund holding Treasuries of twenty years and longer, in dollars.

On 23 July 2026 the yield closed at 4.70, up from 4.67 the day before, while the fund closed at $83.17, down from $83.44. Both lines are "closing" figures, so a chart that goes up means opposite things: rising yields, falling prices. Check which one you are reading before you read the direction.

Try it now

1. Price a 2-year, $1,000 face, 4% annual coupon bond at a 6% yield: 40 ÷ 1.06 + 1,040 ÷ 1.06² = 37.74 + 925.60 ≈ $963.33. Then price the same bond at 2%: 39.22 + 999.62 ≈ $1,038.83. Feel the seesaw under your own pen.
2. Compare the size of those two moves against the 3-year example above. Shorter bond, smaller swing — that's duration, without the word.
3. Go back to the two charts in the section above and find 2022 on both. The yield climbed through the year while the fund fell. Read the size of each move, and check the direction against the arithmetic you just did by hand.