Bond Pricing, Yield, and Duration Explained

Introduction

A bond is a contract with a fixed payment schedule, and almost every technical question you will get about bond math follows from that single fact. The issuer promises a set of coupon payments on known dates plus the return of principal at maturity. Those cash flows are locked in at issuance. When the market's required return changes, the payments cannot move, so the price has to.

That is the entire logic behind the price/yield relationship, and it is why interviewers in debt capital markets, leveraged finance, and markets roles keep returning to the same handful of questions. "What happens to a bond's price when rates rise?" "What is duration?" "Why does a 10-year bond move so much more than a 2-year?" None of these are trick questions. They test whether you understand a present value calculation well enough to reason about it out loud, without a spreadsheet.

This post walks through the mechanics in the order they build on each other: how a bond is priced as the present value of its cash flows, why price and yield move inversely, the difference between coupon rate, current yield, and yield to maturity, how a credit spread sits on top of a risk-free benchmark, what Macaulay and modified duration actually measure, and why convexity means duration is only ever an approximation. Every concept is worked through with real numbers, because interviewers routinely ask candidates to estimate a price move in their heads.

A bond's price is not set by a formula the issuer chooses. It is the number that makes the present value of the promised cash flows equal to what a buyer is willing to pay today, given the return that buyer demands. Every bond pricing question in an interview is a discounting exercise wearing different clothes.

The Three Inputs That Set the Price

Three things determine the price of a plain vanilla bond: the size and timing of the coupons, the principal repayment at maturity, and the discount rate the market applies to those payments. The first two are contractual and fixed. The third is not fixed at all, and it moves every day with the level of interest rates, the issuer's credit quality, and investor appetite for risk.

Written formally, the price of a bond paying an annual coupon $C$ for $n$ years, repaying face value $F$ at maturity, discounted at yield $y$, is:

The first term is an annuity: the stream of coupons. The second term is a single lump sum: the principal. That is the whole model. Everything else in bond math, including duration and convexity, is a statement about how this expression behaves when $y$ changes.

Running the Present Value Math

Take a $1,000 face value bond with a 4% annual coupon and five years to maturity. If the market demands a 5% return on bonds of that maturity and credit quality, you discount $40 in each of years one through four and $1,040 in year five (the final coupon plus principal) at 5%:

That works out to $956.71. The bond does not trade at $1,000 because a buyer receiving only $40 a year on a $1,000 claim, when the market pays 5%, will not pay full face value. The $43.29 discount to par is precisely the compensation that lifts a 4% coupon stream up to a 5% total return.

Two conventions are worth knowing before an interview. US Treasuries and US corporate bonds pay semi-annual coupons, while eurozone sovereigns and most Eurobonds pay annually, so the real calculation on a semi-annual bond uses half the annual coupon discounted over twice as many periods at half the annual yield. And bond prices are quoted as a percentage of face value, so the bond above is quoted at 95.671, not $956.71. Interviewers will not usually penalize you for working in annual periods to keep the mental math clean, provided you say that is what you are doing.

The inverse relationship between price and yield is the single most commonly asked bond question, and the answer that scores is not "because they just do." It is a statement about fixed cash flows meeting a changing discount rate.

The Coupon Is Fixed, So the Price Adjusts

Imagine you own the 4% bond described above and, the day after you buy it at par, newly issued bonds of identical maturity and credit quality come to market with 5% coupons. Nobody will pay you $1,000 for a 4% stream when they can get 5% elsewhere. The only way your bond clears is if its price falls far enough that the buyer's total return, coupons plus the pull toward par at maturity, matches the 5% available on new paper.

Mechanically, the same thing shows in the formula: every term in the present value expression has $(1+y)$ in the denominator. Raise $y$ and every term shrinks. Nothing in the numerator can move to offset it, because the numerators are contractual promises.

A Worked Example: Rates Rise 100 Basis Points

Using the five-year bond, a move in required yield from 4% to 5% takes the price from $1,000 to $956.71, a decline of 4.33%. Run it the other way: if yields fall to 3%, the price rises to $1,045.80, a gain of 4.58%. Notice that the gain when yields fall is slightly larger than the loss when yields rise by the same amount. That asymmetry is convexity, and it comes up later in this post.

The mirror-image logic is why bond investors talk about rate moves in basis points. One basis point is one hundredth of a percentage point, and on large positions a single basis point is real money. The Treasury's daily yield curve data is the reference point most desks quote against: on July 22, 2026 the 2-year traded at roughly 4.31%, the 10-year at 4.67%, and the 30-year at 5.15%.

Par, Premium, and Discount

Once you accept that price adjusts to deliver the market's required yield, the three price states follow immediately:

That convergence has a name: pull to par. Whatever a bond's price today, absent default it repays exactly face value at maturity, so premium bonds drift down and discount bonds drift up as the maturity date approaches. This is also why a bond's price volatility declines as it ages; the remaining cash flows sit closer in time and are discounted less severely.

Three numbers are all described as a bond's "yield," they are almost never equal, and mixing them up is one of the fastest ways to lose credibility in a debt-focused interview.

Coupon Rate: The Promise, Not the Return

The coupon rate is the annual coupon divided by face value, fixed at issuance and printed in the indenture. A $1,000 bond with a 4% coupon pays $40 a year for its entire life, whether it trades at 85 or 110. The coupon rate tells you what the issuer owes. It tells you nothing about what a buyer in today's market will earn.

Current Yield: Income Only, No Capital Gain

Current yield divides the annual coupon by the current market price:

For the discounted bond above, that is $40 divided by $956.71, or 4.18%. Current yield is a useful income snapshot for an investor who cares about cash coming in this year, and it is genuinely used by income-focused buyers. But it ignores the $43.29 capital gain waiting at maturity, so it understates the return on a discount bond and overstates it on a premium bond.

Yield to Maturity: The Number the Market Quotes

Yield to maturity is the single discount rate that sets the present value of all remaining cash flows equal to the current market price. It is an internal rate of return, so there is no closed-form solution for a multi-period bond; a calculator or spreadsheet solves it iteratively. Practically, YTM is the number desks quote, the number index providers aggregate, and the number that shows up as the market-based cost of debt when you build a WACC calculation using traded debt rather than the coupon on the balance sheet.

Yield to Maturity (YTM)

Yield to maturity is the total annualized return an investor earns by buying a bond at its current market price and holding it until maturity, assuming every coupon and the principal are paid in full. It captures both the coupon income and any gain or loss from the difference between the purchase price and face value, which is why it differs from the coupon rate whenever a bond trades away from par.

Laid out side by side for the five-year bond priced at $956.71, the three measures diverge in a predictable direction:

For a discount bond, coupon rate is below current yield, which is below YTM. For a premium bond the ordering reverses exactly. If you can state that ordering rule and explain why, you have answered the question properly.

Yield to Call and Yield to Worst

Many corporate bonds, particularly in high yield, are callable after a set non-call period. If the issuer can redeem early at a specified price, holding to maturity is no longer the only scenario. Yield to call runs the same IRR calculation to the first call date and call price instead of maturity, and yield to worst is simply the lowest of the yields across all possible redemption dates. High yield investors quote yield to worst by default because assuming the issuer will not call a bond trading well above the call price is wishful thinking.

Bond math is the fastest thing to fumble under interview pressure: Work through pricing, yield, and duration questions with full worked answers, start practicing interview questions for free and find the gaps before a DCM interviewer does.

Everything above assumed a single discount rate without asking where it comes from. For corporate bonds it comes from two pieces stacked on top of each other, and interviewers like this question because it forces you to separate interest rate risk from credit risk.

Building a Corporate Yield From the Benchmark Up

A corporate bond yield is the yield on a comparable-maturity government benchmark plus a credit spread that compensates the investor for default risk, illiquidity, and any structural features of the instrument:

If the 10-year Treasury yields 4.67% and a solid BBB rated issuer prints a new 10-year at a spread of 120 basis points, the bond yields 5.87%. That spread is what the market charges this specific borrower for the possibility that it does not pay. The benchmark component reflects the level of rates set in large part by monetary policy, which is why bond desks watch every Fed decision closely; as of its June 2026 meeting the FOMC held the federal funds target at 3.50% to 3.75%.

Credit Spread

A credit spread is the additional yield a corporate bond pays above a government bond of similar maturity, quoted in basis points. It is the market's price for the issuer's default risk and the bond's relative illiquidity. Spreads widen when investors grow more risk averse and tighten when credit conditions improve, often moving independently of the underlying level of interest rates.

Where Spreads Sit in 2026

Spreads are the clearest live indicator of credit market sentiment. Through mid-2026 they have been notably tight by historical standards: the ICE BofA US High Yield Index option-adjusted spread sat around 277 basis points in July 2026, while broad investment grade index spreads have traded inside 100 basis points for much of the year. For context on why that gap exists at all and how the two markets differ in covenants, buyer base, and default experience, the comparison of investment grade and high yield bonds covers the structural differences behind the number.

Two forces move a corporate bond's price, then, and they can offset each other. Treasury yields can rise while spreads tighten, leaving the all-in yield roughly unchanged. Being able to decompose a move into its rate and credit components is exactly the thinking a leveraged finance desk does on every deal, because a widening in high yield spreads can shut a financing that a rate move alone would not.

Duration is the concept interviewers use to separate candidates who memorized a definition from candidates who understand what a present value calculation does. The word is used for two related but distinct measures, and being clear about which one you mean is half the answer.

Macaulay Duration: Weighted Average Time to Get Paid

Macaulay duration is the weighted average time until a bondholder receives the bond's cash flows, where each date is weighted by the present value of the cash flow arriving on it, divided by the total price:

It is measured in years. For a zero coupon bond, Macaulay duration equals maturity exactly, because there is only one cash flow and it arrives at the end. For a coupon bond, duration is always shorter than maturity, because some cash is returned along the way. A 5-year bond with a 4% coupon priced at par has a Macaulay duration of about 4.63 years: the coupons pull the average payment date forward by roughly four and a half months.

Modified Duration: The Price Sensitivity Number

Modified duration converts that time measure into a sensitivity measure by dividing by one plus the periodic yield, where $k$ is the number of coupon periods per year:

On an annual-pay bond that is simply Macaulay duration divided by one plus the annual yield; on a semi-annual bond you divide by one plus half the annual yield. For our 5-year annual-pay bond, that is 4.63 divided by 1.04, or 4.45. The interpretation is direct: for a 100 basis point change in yield, the bond's price changes by approximately 4.45% in the opposite direction. The general approximation is:

Modified Duration

Modified duration measures the approximate percentage change in a bond's price for a 1% (100 basis point) change in yield. A modified duration of 7 means the bond loses about 7% of its value if yields rise 100 basis points and gains about 7% if yields fall 100 basis points. It is the standard measure of interest rate risk for individual bonds and for entire fixed income portfolios.

A Worked Duration Calculation

Test the approximation against the exact price. The 5-year 4% bond has modified duration of 4.45, so a 100 basis point rise should cost about 4.45% of $1,000, giving a predicted price of roughly $955.50. The exact present value calculation earlier produced $956.71. The estimate was off by about $1.21, or roughly a tenth of a percent, and it was conservative: the actual loss was smaller than duration predicted.

That gap is not an error in the arithmetic. It is convexity, and it grows with the size of the yield move and the length of the bond.

Two related measures come up on trading desks. Dollar duration expresses sensitivity in currency terms rather than percentages: modified duration multiplied by market value, which is the currency change for a 100 basis point move. Scaled down to a single basis point, the same idea is quoted as DV01, the dollar value of an 01. Portfolio managers care about DV01 because it aggregates cleanly across positions of different sizes, whereas percentage durations have to be weighted before they can be added.

This is the classic follow-up, and the answer is not "because it is longer." The answer is about where the cash flows sit on the timeline and how compounding treats distant payments.

Where the Cash Flows Sit on the Timeline

Discounting compounds. A cash flow arriving in year 10 is divided by $(1+y)^{10}$, so a change in $y$ hits it far harder than the same change hits a payment arriving in year two. For a 2-year bond, the bulk of the value is the principal repayment arriving very soon, which barely reprices. For a 10-year bond, the principal is eight years further out and takes the full force of the compounding.

Put differently, the 2-year holder gets their money back quickly and can reinvest at the new higher rate almost immediately. The 10-year holder is locked into a below-market coupon for another decade, and the price has to fall enough today to compensate a buyer for that.

A Side-by-Side Rate Shock

Compare three bonds, each with a 4% annual coupon and each priced at par when yields are 4%, and move yields to 5%:

The 10-year loses roughly four times as much value as the 2-year on an identical 100 basis point move, and the 30-year loses roughly eight times as much. This is why a bond fund's stated average duration tells you more about its risk than the credit quality of its holdings during a rate shock, and why insurers and pension funds, whose liabilities stretch decades, deliberately buy long duration assets to match them.

What Else Moves Duration Up or Down

Maturity is the biggest driver, but three other factors matter and interviewers occasionally probe them:

Duration describes a straight line. The actual relationship between price and yield is a curve. Convexity is the name for that curvature, and it explains every gap between the estimates and the exact prices in the table above.

Why the Duration Estimate Drifts

Because duration itself changes as yields change, a single duration number can only be accurate for a small move. As yields rise, duration falls, so the bond loses value at a decelerating rate. As yields fall, duration rises, so the bond gains value at an accelerating rate. The result is that a positively convex bond gains more when yields drop than it loses when yields rise by the same amount.

The 30-year row makes the point starkly. Duration predicted a 17.3% loss and the true loss was 15.4%, a difference of just over $19 per $1,000 bond. Adding a second-order term fixes most of the gap:

Here $\text{Cx}$ is the bond's convexity, not the coupon. Because the yield change is squared, the sign of that second term follows the sign of convexity rather than the direction of the yield move, and for a standard bond convexity is positive, so the term is always a benefit. That is why convexity is described as a benefit an investor owns, and why two bonds with the same duration but different convexity are not equivalent risk.

Convexity

Convexity measures how much a bond's duration changes as yields change, capturing the curvature of the price/yield relationship. Positive convexity means price gains from falling yields exceed price losses from an equal rise in yields, so it works in the bondholder's favor. Bonds with longer maturities, lower coupons, and more dispersed cash flows have higher convexity.

Negative Convexity: Callable Bonds and Mortgages

Not all bonds are positively convex. When an issuer holds an option against the investor, the curvature can invert. A callable bond rallying toward its call price stops appreciating, because the market knows the issuer will redeem it: upside is truncated while downside is not. Mortgage-backed securities behave the same way for a different reason, since homeowners refinance when rates fall, handing principal back to investors precisely when reinvestment rates are worst.

Embedded options cut both ways depending on who owns them. A convertible bond gives the option to the investor rather than the issuer, so its price behavior turns increasingly equity-like as the underlying stock rises rather than being capped. Knowing which side holds the option, and therefore which way the convexity runs, is a strong signal in any credit interview.

These questions come up across every debt-facing seat: debt capital markets, leveraged finance, credit research, and sales and trading. The bar is not computation speed. It is whether you can explain the mechanism and then estimate a number.

The Questions That Come Up Most

Candidates targeting capital markets seats should also expect the surrounding context: how a new issue is priced, marketed, and allocated. The distinction between equity and debt capital markets desks frames where these calculations sit inside the day job, since a DCM banker uses yield and spread language on every call with an issuer.

Mistakes That Get Candidates Cut

One more habit separates strong candidates: sanity-checking direction before computing magnitude. If you are asked what happens to a premium bond as it approaches maturity, the answer is that its price drifts down toward par, and you should be able to say that instantly before touching a formula. Interviewers watch for whether your first instinct is right, because on a desk the direction matters more than the third decimal place.

Get the complete technical reference: Download our comprehensive 160-page PDF, access the IB Interview Guide covering bond pricing, valuation methods, and the full set of technical questions banks actually ask.

Bond math rewards understanding over memorization. Every result above comes from one equation, the present value of a fixed set of cash flows, and every interview question is a different way of asking whether you can reason from it. If you can price a bond, explain why the price moved, estimate the move with duration, and then say why the estimate was slightly off, you have covered the entire topic as it is tested.

Work the numbers by hand a few times before your interviews. Price a five-year bond at three different yields, compute the duration, and check the approximation against the exact price. The arithmetic takes ten minutes and it converts a set of definitions into something you can actually reason with when someone asks you to estimate a price move without a calculator.