Why Do Bond Prices Fall When Interest Rates Rise? Duration and Convexity Explained

Short answer: a bond's coupon is fixed for life, so when market interest rates rise, the only thing that can adjust to bring the bond back into line with what investors now demand is its price — and it adjusts downwards. Duration tells you roughly how far the price moves; convexity tells you why that estimate is always slightly wrong, and always wrong in the bondholder's favour.

This is one of the most reliably asked questions in any debt capital markets, credit or fixed income interview, and it is also one of the most commonly fumbled. Candidates know the direction — prices down, yields up — but stumble when asked why, and then stumble again when asked by how much. This article works through the intuition, the arithmetic and the two follow-up concepts (modified duration and convexity) that separate a memorised answer from a real one. If you want to work the numbers yourself first, the full worked example sits in the case study Bond Basics: Duration, Yield, Price.

The inverse relationship between bond prices and interest rates

A bond is a contract. The issuer promises a fixed stream of coupon payments and a fixed principal repayment at maturity. Neither of those numbers changes when the world changes. If you own a five-year corporate bond with a $1,000 face value and a 5.0% annual coupon, you receive $50 every year and $1,000 at the end, regardless of what happens to central bank policy, inflation expectations or the government bond curve.

What does change is the return investors demand for lending money for five years. That required return is the yield to maturity, or YTM. And because the cash flows are locked, the market can only express a change in required return by changing the price it is willing to pay. That is the entire mechanism behind the inverse relationship between bond prices and interest rates.

A worked example: the $1,000 bond at a 6% yield

Take the bond above — $1,000 face value, 5.0% (0.050) annual coupon, five years to maturity — and suppose the market yield for that credit is now 6.0% (0.060). Pricing the bond means discounting every cash flow at 6.0%:

The bond is worth $957.88, a discount of just over $42 to its $1,000 face value. Nobody will pay par for a 5% coupon when the market is offering 6% on equivalent risk, so the price falls until the total return from here to maturity — the five coupons plus the $42 capital gain as the price climbs back to $1,000 — adds up to 6.0% a year. The discount is not a judgement about the issuer. It is arithmetic.

This is exactly the same discounting logic that underpins a company valuation. If you have worked through what a DCF actually is, you have already met the machinery: project the cash flows, pick a discount rate, sum the present values. A bond is simply the version where the cash flows are contractual rather than forecast, which is why it is such a clean teaching example.

Discount, premium and the pull to par

The relationship works symmetrically. Compare the coupon rate against the market yield and you know immediately where the bond trades:

Whichever side of par the bond starts on, it must converge back to $1,000 by maturity, because that is the amount contractually repaid. Traders call this the pull to par, and it means a discount bond delivers a slow, mechanical capital gain over its remaining life while a premium bond delivers a slow capital loss. A five-year bond has five years for that convergence to work; a bond with six months left has almost none, which is the first clue as to why maturity matters so much for price sensitivity.

Why this is not the same as credit risk

A frequent confusion in interviews is treating a falling bond price as a signal that the issuer is in trouble. It can be, but the two effects are separate. The yield an investor demands has two components: the risk-free rate for that maturity, and a credit spread compensating for the possibility that the issuer does not pay. A price fall driven by a rise in the risk-free rate hits every bond in the market simultaneously, including government bonds with no default risk at all. A price fall driven by a widening credit spread is specific to that issuer.

Distinguishing the two is the daily work of a credit analyst, and it becomes acute when a borrower is genuinely distressed — at which point the conversation shifts from duration to recovery, as in Distressed LBO and Debt-for-Equity Swap. For everything short of that, assume the interviewer asking about bond prices and interest rates wants the rate story, not the credit story.

Duration: putting a number on interest rate sensitivity

Knowing the direction is a starting point. The follow-up question is always "by how much?", and the answer is duration. Confusingly, the word covers two related but distinct measures, and mixing them up is the single most common error in fixed income interviews.

Macaulay duration: the weighted-average wait

Macaulay duration is the weighted-average time until an investor receives the bond's cash flows, where each year is weighted by the share of total present value arriving then. For the bond above it works out at 4.53 years — not the full five, because the coupons return cash along the way and drag the average forward.

What the calculation exposes is how lopsided the weighting is. Year 5 carries $784.62 of the $957.88 total present value, because the $1,000 principal repayment sits there alongside the final coupon. More than 80% of the bond's value is concentrated in a single, distant cash flow. That concentration is the reason duration behaves the way it does.

Modified duration: the number that actually gets quoted

Modified duration converts that time measure into a price-sensitivity measure by dividing by one plus the yield:

Modified Duration = Macaulay Duration / (1 + y) = 4.53 / 1.06 = 4.28

Read it as: this bond loses roughly 4.28% of its value for every one percentage point (100 basis points) rise in yield. On a $957.88 price, that is about $40.98. This is the number a trader, a treasurer or a risk manager quotes, because it answers the question that costs money. Macaulay duration answers a question about time; modified duration answers a question about dollars. If an interviewer asks how much the price moves and you answer "4.53", you have answered the wrong question.

A related quantity you will hear on a desk is DV01 (dollar value of one basis point), which is simply the same sensitivity scaled to a single basis point — here roughly $0.41 per bond. Traders hedge in DV01 terms because it lets them size an offsetting position in a completely different instrument.

What pushes duration up and down

Three drivers matter, and each follows directly from the weighted-average logic:

• Maturity. Longer maturity pushes the principal repayment further out, raising the weighted-average wait. A 30-year bond is dramatically more rate-sensitive than a two-year bond.
• Coupon. A lower coupon returns less cash early, so more weight sits on the final payment. At the limit, a zero-coupon bond pays nothing until maturity, so 100% of its present value is in the final year and its Macaulay duration equals its maturity exactly.
• Yield level. A higher yield discounts distant cash flows more heavily, shifting weight towards the earlier ones and modestly shortening duration.

This is why the phrase "long duration" became shorthand for rate exposure across markets far beyond bonds. Analysts describe a high-growth technology company as a long-duration asset for exactly this reason: most of its value sits in distant cash flows, so it reprices hard when discount rates move. The mechanism is identical to why terminal value dominates a DCF — value concentrated far out in time is value that is highly sensitive to the rate used to discount it.

Convexity: where the duration estimate breaks down

Duration draws a straight line through a relationship that is genuinely curved. The true price-yield relationship for a bond is convex, meaning it bends. For small yield moves the line and the curve are effectively identical, which is why duration survives as a shorthand. For larger moves, the gap opens up.

The asymmetry that favours the bondholder

Reprice the same bond directly at 5.0% and 7.0% and compare against what modified duration predicted:

The error is not random. Duration overstates the loss when yields rise and understates the gain when they fall. A bondholder does better than the linear estimate in both directions. That asymmetry is positive convexity, and it is a genuine economic benefit — which is why investors will accept a slightly lower yield on a more convex bond, and why convexity is priced rather than ignored.

Negative convexity: callable bonds and mortgage securities

The favourable asymmetry is not universal. Callable bonds and mortgage-backed securities exhibit negative convexity. When yields fall, the issuer refinances or homeowners prepay their mortgages, so the investor never captures the full price appreciation — the upside is capped precisely when it would have been most valuable. When yields rise, the call option is out of the money and the investor absorbs the full loss.

That is the mirror image of the pattern above, and it explains why callable paper must offer a higher yield than an otherwise identical bullet bond. The investor has effectively sold an option to the issuer and needs paying for it. The same instinct — who holds the optionality, and what is it worth — runs through the capital structure decisions in LBO debt structures, where call protection, PIK toggles and prepayment terms are negotiated precisely because they shift value between borrower and lender.

When convexity actually matters

For a 25 basis point move, the convexity adjustment on a five-year bond is pennies and nobody mentions it. For a 300 basis point repricing of the kind seen in 2022, it becomes the entire conversation, and it matters more the longer the duration and the more curved the instrument. The practical rule: quote duration for the first-pass answer, then flag that it is a linear approximation and that convexity works in your favour on a vanilla bond. Interviewers are listening for that second sentence.

Why interviewers keep asking this

The question survives because it tests three things at once with a single prompt. First, whether you understand present value well enough to see that a fixed cash flow stream must reprice when the discount rate changes. Second, whether you can quantify a risk rather than merely describe it. Third, whether you know the limits of your own tool — the candidate who volunteers that duration is an approximation is signalling a different level of understanding than the one who does not.

It is also genuinely load-bearing knowledge. Interest rate risk is what determines whether a leveraged borrower can still service its debt, which is the heart of Debt Capacity, and it is what a treasurer manages when deciding between fixed and floating funding. It shows up again in the amortisation and cash sweep mechanics of an LBO debt schedule, where the interest rate assumption drives every line.

How bond mechanics connect to valuation and deal work

The most direct link is the cost of debt in a weighted average cost of capital calculation. A company's true borrowing cost is the market yield on its outstanding debt, not the coupon printed on bonds it issued in a different rate environment. Using the coupon is a classic error, and it flatters the WACC of any company that borrowed cheaply years ago. If that distinction is not yet second nature, the WACC explainer and the practice case WACC: The Building Blocks cover it properly.

The same mechanics reach into equity capital markets too. Where a company sits on the rate cycle affects both what it can borrow and what it can raise in equity, which is one reason issuance windows open and close as fast as they do — a dynamic visible throughout The IPO Process A-Z. And when the borrower is a foreign subsidiary, the risk-free rate itself becomes a question, which is where International WACC picks up.

Common misconceptions, corrected

Key takeaways

• Bond cash flows are fixed by contract, so a change in required yield can only be expressed through the price — hence the inverse relationship.
• A bond trades at a discount when its coupon rate is below the market yield, at a premium when it is above, and converges to par at maturity regardless.
• Macaulay duration measures the weighted-average time to repayment in years; modified duration measures the percentage price change per one percentage point yield move. Quote the second when asked about price.
• Duration is linear, the real relationship is curved, and convexity is that curvature. On a vanilla bond it makes losses smaller and gains larger than duration predicts.
• Callable bonds and mortgage-backed securities have negative convexity and must pay a higher yield in compensation.

The fastest way to make this stick is to price a bond by hand once, then reprice it 100 basis points higher and see the gap between what duration predicted and what actually happened. That is exactly the sequence in Bond Basics: Duration, Yield, Price, and the step-by-step arithmetic is laid out in the companion walkthrough on how leverage and cash flow interact under different rate assumptions.