Bond pricing and yield: why prices fall when rates rise
A bond is a stream of fixed payments, so its price is a present value, and that value moves against the interest rate. This lesson prices a bond from its coupon and yield, separates the three yields people confuse, and uses Apple's debt note to show how fair value can sit well below the amount owed.
- Level
- Beginner
- Reading time
- About 27 minutes
- Know this first
- Annuities, perpetuities and loans
By the end you will be able to:
- price a bond as the present value of its coupons and face value
- explain why bond prices and yields move in opposite directions
- tell the coupon rate, current yield and yield to maturity apart
- read a company's debt note and explain why fair value differs from the amount owed
A bond is a loan you can buy and sell
When a company or government needs to borrow a large sum, it often issues bonds: it divides the loan into thousands of equal pieces and sells them to investors. Each bond is a promise. The issuer will pay a fixed amount of interest at regular dates, and will repay a stated sum on a stated date. A few terms describe it.
- The face value (also called par or principal) is the amount repaid at the end. This lesson uses $1,000.
- The coupon is the fixed interest paid each year, as a percentage of face value. A 4% coupon on a $1,000 bond pays $40 a year, usually in two halves.
- The maturity is the date the face value is repaid.
- The price is what the bond sells for in the market today. It can be above or below face value.
- The yield is the return an investor earns if they buy at that price and hold to maturity.
The coupon is fixed for life. The price is not. After issue, bonds are bought and sold every day, and the price moves with interest rates in the market. This lesson shows why, and it is a direct application of the present value formula.
The price of a bond
A bond is two streams of cash. One is the coupons, a level annuity. The other is the face value, a single payment at maturity. The price is the present value of both, discounted at the yield the market now demands for bonds of this kind and this risk. With coupons twice a year, the discount rate per half-year is the yield divided by 2, and the number of periods is twice the years.
If the market yield equals the coupon rate, the price is exactly the face value: the bond pays what the market wants and no more. If the market yield is higher than the coupon, the bond is less attractive than new bonds, and its price falls below face value until its return matches. If the market yield is lower, the price rises above face value. Prices and yields move in opposite directions, like a see-saw.
Price a bond and move the yield
Set the bond's coupon, the market yield and the years left to maturity. Face value is fixed at $1,000, and coupons are paid twice a year. The left page shows the price, how far it is from face value and the cash coupon. The right page shows the current yield, then what happens to the price if market yields rise or fall by one point, and the bond's sensitivity to that move. The numbers are an illustration with made-up terms.
A 4.00% bond with 10 years left, priced to yield 5.00%, is worth $922 per $1,000 of face value, at a discount. The coupon is $40 a year, which is 4.34% of the price. If yields rose one point the price would fall to $851; if they fell one point it would rise to $1,000.
Four experiments. First, set the yield equal to the coupon, both at 4%. The price is $1,000, exactly face value. Now move the yield to 3%: the price rises to $1,086. Move it to 6%: the price falls to $851. The see-saw is there. Second, set the coupon back to 4% and the yield to 5%, and compare 2 years with 30 years to maturity. The 2-year bond falls only to $981, while the 30-year bond falls to $845. Long bonds are far more sensitive to rates. Third, set the coupon to 0%, a zero-coupon bond, with 10 years at a 5% yield. The price is $610, and a one-point rise in yield cuts it by 9.3%, a bigger fall than for the 4% coupon bond, because all of the return comes at the end. Fourth, set one year to maturity: the price is close to face value whatever the yield, since there is little time for rates to matter.
How price moves against yield
The table gives the price of a 4% coupon bond across market yields and lives. Read down a column to see the effect of time, and across a row to see the effect of yield.
| Years to maturity | Yield 2% | Yield 3% | Yield 4% | Yield 5% | Yield 6% | Yield 8% |
|---|---|---|---|---|---|---|
| 2 years | $1,039 | $1,019 | $1,000 | $981 | $963 | $927 |
| 10 years | $1,180 | $1,086 | $1,000 | $922 | $851 | $728 |
| 30 years | $1,450 | $1,197 | $1,000 | $845 | $723 | $548 |
Take the 10-year bond. A move in yield from 4% to 6%, two points, takes the price from $1,000 to $851, a fall of 14.9%. Moving from 4% to 2% raises it to $1,180. The gains and losses are not symmetric: a fall in yield raises the price by more than a rise of the same size lowers it. This curvature is called convexity, and it works in the holder's favour.
Yield to maturity
The price formula can be turned round. Given a price in the market, the yield to maturity is the single discount rate that makes the present value of all the remaining coupons and the face value equal to that price. It is the bond's internal rate of return, an idea the next lesson develops. It has no neat formula, and is found by trial and error, which is what a calculator does.
Take the 4% coupon bond with 10 years to run that sells for $922. Trying yields until the present value equals the price gives a yield of 5.00%, which is what we started with. Three yields are often confused.
- Coupon rate: the fixed interest as a share of face value. It never changes.
- Current yield: the yearly coupon divided by today's price. For this bond, $40 ÷ $922 = 4.34%. It ignores the gain from the price rising back to $1,000 at maturity.
- Yield to maturity: the full return from holding to the end, counting coupons, the repayment and the gap between price and face value. It is the figure to compare across bonds.
The ranking is always the same for a bond at a discount: coupon rate below current yield below yield to maturity. For a bond at a premium the order reverses.
Duration: a measure of sensitivity
The widget reports a figure called duration. As used here it is the modified duration: the approximate percentage change in price for a one percentage point change in yield. A duration of 8.1 years means that a one-point rise in yield lowers the price by about 8.1%. It is estimated in the widget from the prices one point above and below. Duration grows with maturity and falls with the coupon, which is why long, low-coupon bonds are the most sensitive and short, high-coupon bonds the least. It is the first thing a bond investor checks, because it says how much a rate change will hurt.
Apple's bonds: the debt note
Apple borrows by issuing bonds, which it calls term debt, as well as short-term commercial paper. The debt note in its fiscal 2025 annual report (the year ended 27 September 2025), in millions of dollars, shows how a balance sheet figure and a market value can differ.
| Apple term debt, 27 September 2025 | Millions of US dollars |
|---|---|
| Principal outstanding | 91,281 |
| Unamortized discount and issuance costs | (309) |
| Hedge accounting fair value adjustments | (294) |
| Carrying amount on the balance sheet | 90,678 |
| of which due within a year | 12,350 |
| of which due later | 78,328 |
| Fair value of the notes (Level 2) | 80,400 |
| Principal due in fiscal 2026 | 12,393 |
| Principal due in fiscal 2027 | 10,078 |
| Principal due in fiscal 2028 | 9,300 |
| Principal due in fiscal 2029 | 5,235 |
| Principal due in fiscal 2030 | 4,972 |
| Principal due after fiscal 2030 | 49,303 |
Apple owes $91,281 million of principal in term debt. After small adjustments for discounts, issue costs and hedging, the amount on the balance sheet is $90,678 million. But the note also gives the fair value of these notes: $80.4 billion. That is the price at which they would trade, the present value of the remaining payments at today's market yields. It is $10,881 million below the amount owed, or 88.1% of principal.
The see-saw explains the gap. Apple's notes were issued over many years, at effective interest rates that run from 0.03% to 5.75%. A large part was borrowed when rates were very low. Apple's own notes issued in 2025 carried effective rates of 4.07% to 4.83%, so today's market asks for far more than the old notes pay. Apple's discussion of its liquidity says that interest on the notes in the next 12 months is $2.6 billion, about 2.8% of principal, a rough figure from rounded numbers. A bond paying about 2.8% of face value, set against a market yield near 4.5%, has to sell below face value.
The widget can show the size of the effect. Take a bond with a coupon of 2.85%, about nine years to maturity and a market yield of 4.5%. Its price is $879 per $1,000, or 87.9% of face value, close to the 88.1% ratio in the note. This is an illustration, not a model of Apple's debt, which has many bonds with different coupons and dates. It shows that the difference between face value and fair value in the note is what the see-saw predicts.
Two more facts from the report give the same picture. Over the life of the notes Apple will pay about $37.0 billion in interest, which is 41% of the principal. And its commercial paper, which matures in under nine months, carried a weighted-average rate of 4.19% at the end of fiscal 2025, down from 5.00% a year earlier. Short rates fell during the year, and the price of short debt barely changes with them.
Does the gap mean Apple has made a gain? Not in the accounts. Notes are carried at amortized cost, about what was borrowed, and Apple still owes the full principal at maturity. The fair value is shown for information. It would matter only if Apple bought the bonds back, which it is free to do but is not obliged to. The fixed rates, in effect, protect it from the rise in market yields. The report adds that a 100 basis point (one percentage point) rise in rates would increase its annual interest expense on term debt by $129 million, 0.14% of principal, which is small against the principal and consistent with most of the notes paying a fixed rate.
What bond prices cannot tell you
- Credit risk is not in the formula. The yield used is the market's, and it includes a premium for the chance the issuer does not pay. A bond from a weaker issuer has a higher yield and so a lower price.
- Many bonds can be repaid early. A callable bond lets the issuer repay before maturity, which limits the price rise when yields fall.
- Duration is a first approximation. For a large move in yield, convexity matters and the real price change differs from the duration estimate.
- Fair value of a debt note is an estimate. It is based on observable prices for similar notes, not on every bond actually trading on the day.
- A real bond has dates. Coupons are paid on set days, and the price quoted may exclude interest that has built up since the last coupon. This lesson assumes you buy on a coupon date.
Coupon dates and yield conventions
A bond price is the present value of its coupons and its face value, and the discount rate is the yield. Two conventions change the number. The first is how often coupons are paid: US corporate bonds usually pay semiannually, while many bonds issued in European markets pay once a year. The second is how the yield is quoted. A yield on a semiannual bond is conventionally stated as a nominal annual rate, so a yield of 5% means 2.5% every half year, and the effective annual yield is 5.062%.
The effect on price is small but real. A 10-year bond with a 4% coupon and a yield of 5% is worth 922.78 per 1,000 of face value if it pays annually and 922.05 if it pays semiannually, a difference of 0.73. Dealers handle it by specifying the convention for each market. An investor comparing yields from two countries has to put them on the same basis before comparing.
How far a price moves
The table prices a 10-year 4% bond with semiannual coupons at four yields. At a yield of 4% the bond is worth its face value, and each point of yield moves the price by roughly seven to nine percent in the direction opposite to the yield.
| Yield to maturity (semiannual) | Price per 1,000 face value | Against face value |
|---|---|---|
| 3% | 1,085.84 | 8.6% |
| 4% | 1,000.00 | 0.0% |
| 5% | 922.05 | −7.8% |
| 6% | 851.23 | −14.9% |
A one-point rise in yield from 4% to 5% cuts the price by −7.8%. On a holding of 1 billion of such bonds, that is a loss of roughly 78 million in market value without any default. This effect is why companies that hold bond portfolios, or that have debt at fixed rates, are exposed to changes in interest rates even when nothing about their own business has changed.
Corporate borrowing in context
The three companies in this module's examples borrow in very different amounts: Apple has $98,657 million of borrowings, Microsoft $43,151 million and Toyota ¥38,792,879 million, which includes the debt of its finance arm. Each figure is the sum of many separate bonds and loans with their own coupons, maturities and currencies, and the notes to the accounts list them. The market price of that debt moves with yields in exactly the way the table shows. The balance sheet usually carries borrowings at amortised cost, with the market value disclosed in the notes, so the two figures can differ when rates have moved since the bonds were issued.
Compare yields only within the same currency. A yen bond and a dollar bond of similar quality have different yields mainly because interest rates and expected inflation differ between the two currencies, and the difference does not mean one is a bargain.
Common mistakes
Thinking a bond's price is fixed once issued
Only the coupon and the repayment are fixed. The price changes every day with market yields.
Confusing coupon and yield
The coupon is a fixed percentage of face value. The yield depends on the price paid. The two are equal only when the price is par.
Assuming a bond is safe because it pays fixed interest
Held to maturity the payments are fixed, but sold early the price may be lower. A 4% coupon, 30-year bond bought at par loses 28% of its value if yields rise to 6% and stay there.
Comparing bonds of different lives by yield alone
A long bond and a short one at the same yield have very different risks. Compare duration as well.
Reading fair value as a loss or a gain
The $10,881 million gap on Apple's notes is a market valuation, not money Apple has earned or lost, as long as it holds the notes to maturity.
Try it yourself
1. A bond has a 5% coupon and sells at a yield of 5%. What is its price per $1,000, and why?
Show the working
$1,000. When the market yield equals the coupon, the present value of the payments equals the face value, so the bond sells at par.
2. A 3-year bond with a $1,000 face value pays a 6% coupon once a year. The market yield is 8%. Find the price.
Show the working
Coupons of $60 are discounted at 8%: 60 ÷ 1.08 = 55.56, 60 ÷ 1.082 = 51.44, and the final year's 1,060 ÷ 1.083 = 841.46. The price is $948.46, below face value because the coupon is below the market yield.
3. Market yields rise. Which falls more in price, a 2-year bond or a 20-year bond with the same coupon, and why?
Show the working
The 20-year bond. Its face value is repaid far in the future and is discounted for many more periods, so a higher yield cuts its present value by more. The calculator above shows this for 2 and 30 years: $981 and $845 at a 5% yield for a 4% coupon.
4. Apple's term debt has principal of $91,281 million and a fair value of $80.4 billion. Is the fair value above or below principal, and what does it say about market yields compared with the coupons on the notes?
Show the working
Below, at 88.1% of principal. A bond sells below face value when market yields are above its coupon. So, on average, the market yield for Apple's notes is above the rates the notes pay.
Where this is used
- A bond fund manager measures the duration of the portfolio to see how much it will gain or lose if rates change.
- A company reports the fair value of its debt in the notes to its accounts, as Apple does.
- A saver buying a bond compares yields to maturity, not coupons, to compare offers.
- A treasurer decides whether to issue fixed-rate or floating-rate debt after looking at how the price would respond to rate changes.
Key terms
- Bond
- A tradeable loan: a promise to pay fixed interest and repay face value on a stated date.
- Face value
- The amount repaid at maturity, also called par or principal.
- Coupon
- The fixed interest a bond pays, stated as a percentage of face value.
- Yield to maturity
- The single discount rate that makes the present value of a bond's payments equal to its price.
- Current yield
- The yearly coupon divided by the bond's price.
- Par, premium, discount
- A bond sells at par when its price equals face value, at a premium above it and at a discount below it.
- Duration
- A measure of how sensitive a bond's price is to a change in yield.
- Fair value
- The price at which an asset or a liability could be exchanged between willing parties, here the market value of a company's notes.
- Term debt
- Apple's name for its longer-term borrowing in bonds, as distinct from short-term commercial paper.
Questions people ask
Why do bond prices fall when interest rates rise?
New bonds pay the higher rate, so older bonds with lower fixed coupons must sell for less for a buyer to earn the same return. The price falls until the yield on the old bond matches the market.
Can I lose money in a bond?
Yes. If you sell before maturity at a lower price, or the issuer fails to pay, you lose. If you hold a safe bond to maturity you receive the stated payments.
What is a zero-coupon bond?
A bond with no coupons that sells at a discount and pays face value at maturity. All the return is the price rising to par, which makes it the most sensitive to yields for its life.
Why does Apple report fair value of debt if it does not sell it?
Accounting rules require companies to disclose the fair value of financial instruments carried at cost, so that a reader can see how far market conditions have moved from the recorded figures.
Sources and review
- Apple Inc., Form 10-K for the fiscal year ended 27 September 2025, filed with the US Securities and Exchange Commission. Source of the term debt principal, future interest and the next 12 months' interest in the discussion of liquidity and capital resources, the commercial paper rates, and the interest rate sensitivity.
- EDGAR Tools, Apple long-term debt disclosures, a data site that reproduces the debt note, used to read the note tables: carrying amount, fair value, effective interest rates and principal due by year.
- ACCA, Bond valuation and yields, for bond pricing and yield to maturity.
- AccountingTools, How to calculate present value, for the present value basis.
- Brilliant, Time value of money, for the relationship of price and discount rate.
- Apple Inc., Form 8-K, Exhibit 99.1: fourth quarter and full-year fiscal 2025 results, filed with the US Securities and Exchange Commission. Source of Apple's statements.
- Microsoft Corporation, Form 8-K, Exhibit 99.1: fourth quarter and fiscal year 2025 results, filed with the US Securities and Exchange Commission. Source of Microsoft's statements.
- Toyota Motor Corporation, FY2025 Financial Summary (IFRS, unaudited, in millions of yen), furnished to the US Securities and Exchange Commission on Form 6-K. Source of Toyota's statements.
How the numbers were checked: Apple's debt note was added up by code. The principal due by year sums to the total principal, and principal less the discount and the hedge adjustment equals the carrying amount, which equals the current and non-current portions together. The next-year repayment and the total principal were matched to the figures in the discussion of liquidity. The bond calculator was tested against a separate price formula for more than two hundred combinations of coupon, yield and term, including that the price is exactly face value when the yield equals the coupon, is above it when the coupon is higher, below it when lower, and that the price always falls when the yield rises.