1. What is a bond?

A bond is a debt instrument. When an issuer sells a bond, it is borrowing money. The buyer of the bond is the lender. In exchange for the upfront cash, the issuer promises to make a series of interest payments during the bond’s life and to return the original amount borrowed on a specified future date.

Unlike a bank loan, a bond is a transferable security. Once issued, it can be bought and sold in the secondary market, so the identity of the lender can change many times over the bond’s life. The issuer’s obligation runs to whoever holds the bond at any given moment.

Four terms define any bond:

• the issuer: the entity borrowing the money and promising to repay,
• the face value (also called par value or nominal value): the amount to be repaid at maturity and the reference amount for coupon calculations,
• the coupon: the interest payment, expressed as a percentage of face value,
• the maturity date: the date on which the face value is returned and the bond ceases to exist.

2. Issuers: who borrows in the bond market?

Bonds are issued across a wide spectrum of borrowers, grouped into three broad categories.

2.1 Government bonds

Government bonds are issued by sovereign states to finance public spending. Because a government can tax its citizens and, in some cases, print its own currency, government bonds are usually treated as the closest available proxy for a risk-free asset in that currency. As we will see in a later chapter, another common proxy for the risk-free rate can be found in swap markets with overnight index swaps.

The benchmark government bond markets are:

• US Treasuries: issued by the United States Department of the Treasury, the deepest and most liquid bond market in the world,
• German Bunds: the benchmark for euro-denominated government debt, issued by the Federal Republic of Germany,
• UK Gilts: issued by His Majesty’s Treasury, denominated in sterling,
• Japanese Government Bonds (JGBs): the dominant instrument in yen fixed income markets.

2.2 Corporate bonds

Corporate bonds are issued by companies to raise debt capital in the public markets. Because companies can default on their obligations in a way that governments of developed economies typically do not, corporate bonds carry a credit spread: additional yield above the government benchmark to compensate investors for default risk.

Credit spreads vary with the issuer’s perceived creditworthiness. Rating agencies such as Moody’s, S&P, and Fitch publish credit ratings that summarise this assessment. Bonds rated BBB or above (on the S&P scale) are classified as investment grade. Bonds rated below BBB are classified as high yield (also called speculative grade or, informally, junk bonds). High yield bonds carry substantially wider spreads to reflect the higher probability of default.

2.3 Supranational and agency bonds

Supranational bonds are issued by international organisations such as the European Investment Bank (EIB), the World Bank, or the International Monetary Fund (IMF). These institutions borrow in the capital markets to fund lending to member states and development projects. Their bonds typically carry a credit rating of AAA and trade at a small spread above the corresponding government benchmark.

Agency bonds are issued by government-sponsored entities (GSEs) or state-owned institutions. In the United States, the mortgage agencies Fannie Mae and Freddie Mac are the largest issuers. Agency bonds benefit from an implicit or explicit government guarantee and trade at spreads that are narrow but not zero.

2.4 Bonds vs loans

Neither instrument is inherently superior. The choice is a strategic one, shaped by the nature and time horizon of the project, the size and risk profile of the company, and its willingness to accept external scrutiny.

Bank loans are valued above all for their flexibility. Drawdown and repayment schedules can be tailored to the borrower’s cash flows, early repayment is usually possible without penalty, and terms can be renegotiated over the life of the facility. Loans are also quick to arrange and negotiated confidentially, and they build the lasting banking relationships that give companies access to advisory support and to credit when it is needed.

Bonds, by contrast, offer direct access to institutional investors such as asset managers and insurance companies. Because no repayment falls due before maturity, they leave the issuer greater freedom in managing its cash flow. They typically carry longer maturities than a loan, lighter collateral requirements and more lenient covenants, and they remain available to issuers rated BB+ or below or carrying higher leverage.

In practice the two are complements rather than substitutes. A common structure executes an acquisition quickly with a bridge loan, then refinances it through a bond issue for stability and longer tenor, with term loans and credit lines alongside. The aim is to diversify funding sources rather than depend on any single one.

3. Coupon structures

Not all bonds pay interest in the same way. Three structures are standard.

3.1 Fixed rate bonds

A fixed rate bond pays a coupon that is set at issuance and does not change for the bond’s life. Each coupon payment is the same amount. If a bond has a face value of €1,000, a coupon rate of 4.00%, and pays annually, the holder receives €40 every year until maturity, then €1,040 on the final date (the last coupon plus face value).

Fixed rate bonds are the most common structure in corporate and government debt markets.

3.2 Floating rate notes

A floating rate note (FRN) pays a coupon that resets periodically against a benchmark interest rate. The coupon is typically expressed as the benchmark rate plus a fixed spread, for example EURIBOR (Euro Interbank Offered Rate) plus 0.80%.

Because the coupon tracks current market rates, an FRN’s price stays close to par throughout its life. Its interest rate risk (sensitivity to parallel shifts in yields) is low compared to a fixed rate bond of the same maturity. FRNs are common in the banking sector, where issuers and investors both want instruments whose cost or return moves in line with short-term rates.

3.3 Zero coupon bonds

A zero coupon bond pays no coupon during its life. Instead, it is issued at a discount to face value and redeems at par. The investor’s entire return comes from the price appreciation between purchase and maturity.

A zero coupon bond maturing in five years at €100 might be purchased for €85. The difference between the purchase price and the redemption value represents the accumulated interest over the bond’s life. Because there are no intermediate cash flows, zero coupon bonds are useful instruments for immunising specific future liabilities and are also the mathematical foundation for discount factors, as explained in the Time value of money chapter.

4. Issuance and pricing at par

When a borrower issues a new bond, the investment bank managing the transaction sets the coupon rate at a level that causes the bond to price at or very close to par (face value of 100).

For an investment grade corporate issuer, this requires two inputs:

• the swap rate for the bond’s maturity, which reflects the interbank interest rate at that tenor in the relevant currency,
• the issuer’s credit spread, reflecting the additional yield investors require to compensate for credit risk.

The coupon is set at approximately the sum of these two: swap rate plus credit spread. At that rate, discounting the bond’s cash flows at the same combined rate produces a present value of 100.

In practice, the process involves bookbuilding, in which the bank solicits demand from institutional investors at various yield levels, then sets the final terms based on where sufficient demand exists. The spread over mid-swaps is fixed at launch, and the coupon is then calculated from that spread and the prevailing swap rate on the pricing date.

5. The secondary market

Once a bond is issued and allocated to investors, it trades in the secondary market. For government bonds, this market is highly liquid: large institutional investors, dealer banks and even retail investors trade in size with narrow bid-offer spreads. For corporate bonds, liquidity varies significantly with issuer size, credit quality, and market conditions.

Bond prices in the secondary market move continuously in response to:

• changes in the underlying risk-free rate (the government bond yield),
• changes in the issuer’s credit spread,
• changes in broader market risk appetite.

When market interest rates rise, bond prices fall. When rates fall, prices rise. This inverse relationship is a direct consequence of discounting and is examined in detail in section 9.

6. Clean price, dirty price, and settlement

6.1 Accrued interest

Between coupon dates, a bond earns interest daily. If you hold a bond for three months of a six-month coupon period and then sell it, you have earned three months of accrued interest even though no coupon has been paid yet. The buyer, who will receive the full next coupon, compensates you for those three months of interest as part of the purchase price.

Accrued interest (AI) is calculated as:

The denominator depends on the day count convention used for that bond. Common conventions include:

• Actual/Actual (ICMA): used for most EUR and GBP government bonds. Both the numerator and denominator count actual calendar days,
• 30/360: used for many EUR corporate bonds. Each month is treated as 30 days and each year as 360 days,
• Actual/360: common in money markets and some FRN structures,
• Actual/365: used for UK Gilts and some other sterling instruments.

The day count convention matters: using the wrong one will produce an incorrect accrued interest calculation and therefore an incorrect settlement amount.

6.2 Clean price and dirty price

Bond prices are quoted in two ways, and the distinction is important for settlement.

The clean price is the price quoted in the market, net of accrued interest. It is the number you see on platforms such as Bloomberg or Reuters. Clean prices are quoted this way because they remove the mechanical drift in price caused by the daily accumulation of accrued interest, making it easier to observe and compare changes in the bond’s underlying value.

The dirty price is the actual settlement amount. It is the clean price plus accrued interest:

When you buy a bond, you pay the dirty price. The accrued interest you pay to the seller is recovered when you receive the next full coupon, which covers the entire coupon period including the days before you owned the bond.

On a coupon date, accrued interest is zero and the clean price equals the dirty price.

6.3 Settlement conventions

Bond trades settle on a standard cycle after the trade date. The settlement date is the date on which payment is made and bonds change hands.

Most EUR and GBP government and corporate bonds settle on a T+2 basis: the trade is agreed today and cash and securities are exchanged two business days later. US Treasuries moved to T+1 settlement in May 2024, meaning trades settle the next business day.

Near a coupon payment date, some markets observe an ex-coupon period: a window during which a bond trades without entitlement to the next coupon. An investor who buys during this window will not receive the upcoming coupon, so the accrued interest turns negative for those days. UK Gilts, for example, have a seven-business-day ex-dividend period. For most EUR corporate and government bonds, no such window exists: the bondholder on the coupon date receives the payment and accrual runs continuously from coupon date to coupon date.

7. Pricing a bond: the sum of discounted cash flows

Recall from the chapter Time value of money & discount factors that the value of any financial instrument is the sum of its future cash flows, each discounted to today:

For a bond, $C_i$ is either a coupon payment or the final coupon plus face value, and $\text{DF}(t_i)$ is the discount factor for that payment date, drawn from the zero-coupon yield curve appropriate to the bond’s issuer. This formula gives the dirty price. To obtain the clean price quoted in the market, subtract the accrued interest.

One crucial implication follows immediately: there is no single discount rate for the whole bond. Each cash flow has its own discount factor, specific to its payment date. This is what distinguishes proper bond pricing from the naive approach of applying one average rate to all payments.

8. Worked example: pricing a 5-year government bond

Bond terms: face value €100, annual coupon 3.00%, maturity five years, day count 30/360 (each year fraction is exactly 1.0).

We follow the compound discounting convention used in the bond market, such that:

Zero rates and discount factors from the market:

Cash flows and present values:

The dirty price is €99.796 per €100 face value. Since we price at issuance, on a coupon date, accrued interest is zero and the clean price equals the dirty price.

The price is slightly below par because the coupon rate (3.00%) is marginally below the five-year zero rate (3.05%). Trading at par means that the clean price is equal to the face value. A bond trading below par is said to trade at a discount. A bond trading above par trades at a premium.

9. The price-yield relationship

A bond’s cash flows are fixed at issuance. Market interest rates change every day. When rates change, the discount factors applied to those fixed cash flows change too, and so does the bond’s price.

The relationship is always the same: when interest rates rise, bond prices fall. When interest rates fall, bond prices rise. This is not a rule of thumb but a direct mathematical consequence of discounting.

10. Yield to maturity

10.1 Definition

The worked example in section 8 used a different discount rate for each maturity. In practice, bond prices are almost never quoted that way. They are quoted as a single number called the yield to maturity (YTM).

The YTM is the single constant discount rate that, applied to all of the bond’s future cash flows, produces the bond’s observed market price. It is the value $y$ that solves:

For our five-year bond at a price of 99.796, the YTM solves:

The solution, found numerically, is $y \approx 3.04\%$. This sits slightly below the five-year zero rate of 3.05%, because the YTM is a weighted average across all five maturities and the shorter-dated zero rates (2.60% at year one, 2.75% at year two) pull the average down. If the price were to increase from €99.796 to €100 (bond becomes at par), then the YTM would have to decrease from $y \approx 3.04\%$ to $y = 3\%$.

10.2 YTM as an internal rate of return

The YTM is exactly the internal rate of return (IRR) of the bond investment. It is the single discount rate that equates the upfront cost of the bond to the sum of all future cash flows received. This makes it the standard metric for comparing bonds: if bond A has a YTM of 4.00% and bond B has a YTM of 3.50%, and they carry similar credit risk and maturity, bond A offers a higher return for the same risk.

10.3 What YTM assumes

The YTM calculation bakes in two assumptions that practitioners must keep in mind.

A flat yield curve. By applying a single rate $y$ to all maturities, YTM implicitly assumes the zero curve is flat. In reality, it has a shape. YTM compresses that shape into a single number, which is useful for quotation but loses the curve structure that proper valuation requires.

Reinvestment at the YTM. The IRR interpretation requires that each coupon received during the bond’s life be reinvested at the same rate $y$ for the remainder of the bond’s life. In practice, reinvestment rates are unknown and will almost certainly differ from the YTM. If coupons are reinvested at lower rates, the realised return will fall short of the YTM; if at higher rates, it will exceed it.

These limitations do not make YTM useless. It is the language of the bond market and fluency in it is essential. But they explain why practitioners use zero curves and discount factors for valuation and risk management, reserving YTM as a convenient quotation and comparison tool.

11. PV01: measuring interest rate sensitivity

11.1 What happens when yields move?

A bond’s cash flows are fixed at issuance. What changes every day is the yield at which those cash flows are discounted. A natural first question is: if the yield moves by the smallest amount the market quotes, one basis point (0.01%), how much does the bond’s price change?

To answer this, take the five-year government bond from the worked example in section 8: face value €100, annual coupon 3.00%, yield to maturity 3.04%, dirty price €99.817. Now reprice the same bond at a yield of 3.05% (one basis point higher) and at 3.03% (one basis point lower):

When the yield rises by one basis point, the price drops by €0.046. When the yield falls by one basis point, the price rises by €0.046. This quantity is the change in price for a one basis point move in yield. It is called PV01 (Price Value of a Basis Point, also written as DV01, PVBP or BPV). It is quoted as a positive amount for a bond you hold. The chapter Net present value & basis point value introduces the basis point value of an interest leg, which counts the interest payments only, and compares it with PV01 (section 4.3).

For this bond, $\text{PV01} \approx €0.046$ per €100 face value.

On a real-world holding of, say, €10 million face value, the PV01 scales proportionally. The prices in the table are rounded to three decimals, so the formula uses the unrounded PV01 of €0.04568 per €100 face value:

A one basis point rise in yield on this position reduces its market value by approximately €4,568. A one basis point fall increases it by the same amount. PV01 is expressed in currency terms, which makes it additive across a portfolio: you can sum the PV01 of every bond you hold to get the total sensitivity of the portfolio to a parallel shift in yields.

A payer swap can hedge this fair value risk of a bond that you hold. The chapter Cash flow vs fair value hedging shows how, and what the hedge creates.

11.2 Does the coupon rate matter?

Suppose a second five-year government bond exists, identical in every way except that it pays a 6.00% coupon instead of 3.00%. At the same yield of 3.04%, this bond trades at a premium since its coupon exceeds the market rate. Its price is €113.540.

Repeating the same exercise, repricing at one basis point higher and lower:

In absolute euro terms, the 6% bond has a slightly higher PV01 per unit of face value: €0.050 versus €0.046. But this comparison is misleading, because you pay more for each unit of face value of Bond B. The question that matters to an investor is: for each euro I actually invest, how much am I exposed to interest rate moves?

Expressed as a percentage of the bond’s price, the picture reverses:

Per euro invested, the 3% coupon bond is more sensitive to interest rate moves than the 6% coupon bond. This is a general result: for two bonds of the same maturity, the one with the lower coupon will always have:

• a lower PV01 in absolute terms, because its price is lower and therefore there are fewer euros at risk,
• but a higher sensitivity to yield changes relative to its price, because more of its value is concentrated in the final payment, where discounting has the greatest impact.

The intuition is straightforward. Bond B, with its higher coupon, returns more cash early in its life. Those near-term cash flows are less affected by a change in the discount rate than cash flows that are five years away. Bond A, with its lower coupon, has more of its value concentrated in the final payment (year five), where the impact of discounting is greatest.

The extreme case makes this vivid. A zero coupon bond pays no coupons at all. It concentrates 100% of its value in a single payment at maturity. Its percentage sensitivity is the highest possible for any bond of that maturity.

11.3 PV01 as a formula

There is a direct formula for PV01 that avoids repricing. It relies on modified duration ($D_{Mod}$), which is introduced in the next section:

For Bond A: $4.578 \times 99.817 \times 0.0001 = €0.0457$.

For Bond B: $4.369 \times 113.540 \times 0.0001 = €0.0496$.

These match the repricing results, confirming that modified duration and the direct repricing approach measure the same thing.

12. Duration: formalising the sensitivity

12.1 From PV01 to modified duration

The percentage PV01 from section 11 already captures the rate of change in price relative to a basis point shift in yield. Modified duration is simply this percentage sensitivity, rescaled to express the price change per 1% (100 basis points) move in yield rather than per basis point:

This says: if the yield rises by $\Delta y$ (expressed as a decimal, so 1% = 0.01), the bond’s price falls by approximately $D_{Mod} \times \Delta y$ percent. The minus sign reflects the inverse relationship between yields and prices.

For Bond A, modified duration is 4.578. If the yield rises by 1% (100 basis points), the price falls by approximately 4.578%. For Bond B, modified duration is 4.369. A 1% rise in yield causes a price drop of approximately 4.369%. The lower coupon bond is more sensitive, consistent with everything observed through PV01.

12.2 Macaulay duration: why the coupon matters

Modified duration has a companion called Macaulay duration, which reveals *why* a lower coupon means higher sensitivity.

Macaulay duration is the weighted-average time at which the bond delivers its cash flows, where each weight is the present value of that cash flow as a share of the bond’s total price:

Computing this for both bonds makes the mechanism visible:

Bond A (3% coupon):

Bond B (6% coupon):

The weight column tells the story. Bond A concentrates 88.8% of its value in the year-five payment. Bond B concentrates only 80.4% in year five, spreading the rest across larger coupon payments in years one through four. Because more of Bond B’s value arrives earlier, its weighted-average maturity is shorter: 4.502 years versus 4.717 years.

Macaulay duration therefore answers a precise question: how many years into the future is the economic centre of gravity of this bond? The further away that centre of gravity, the more each basis point move in yield affects the bond’s price, because distant cash flows are more sensitive to changes in the discount rate. This is why lower coupon bonds have longer Macaulay duration, and why longer duration translates directly into higher sensitivity.

A zero coupon bond is the extreme: all value is in a single cash flow at maturity, so its Macaulay duration equals its time to maturity exactly (5 years for a five-year zero). This is the maximum possible Macaulay duration for any bond of a given maturity.

12.3 Connecting the two durations

Modified duration is derived from Macaulay duration by a single adjustment:

For Bond A: $4.717 / 1.0304 = 4.578$.

For Bond B: $4.502 / 1.0304 = 4.369$.

Macaulay duration tells you *where* the cash flows sit in time. Modified duration tells you *how much* the price moves relative to its initial price when yields shift. The division by $(1+y)$ is a technical adjustment that converts a time-weighted average into a first-derivative sensitivity measure. In practice, the two numbers are close. They differ only by the yield scaling and both carry units of years.

12.4 Summary of the three measures

The three are mathematically linked: $\text{PV01} = D_{Mod} \times \text{Price} \times 0.0001$, and $D_{Mod} = D_{Mac} / (1 + y)$. Knowing any one of them plus the bond’s price and yield is enough to derive the other two.

13. Convexity: why PV01 is not constant

13.1 PV01 changes as yields change

The PV01 computed in section 11 applies at a specific starting yield. But PV01 is not constant: it depends on the level of yields. The table below reprices Bond A at different yield levels and computes the PV01 at each:

PV01 falls as yields rise and increases as yields fall. This means the price-yield relationship is not a straight line. If it were, PV01 would be the same everywhere. Instead, the relationship is curved: the bond becomes more sensitive when yields are low and less sensitive when yields are high. This curvature is called convexity.

13.2 The practical consequence

Consider a €10 million face value position in Bond A. Its PV01 is €4,568 per basis point. A simple linear estimate of any yield move multiplies this PV01 by the number of basis points. A 50 basis point rise, for instance, would be estimated as 50 × €4,568 = €228,398 of losses. A 100 basis point rise would be 100 × €4,568 = €456,796. This linear estimate assumes PV01 stays constant regardless of how far yields move.

The table below compares this linear estimate to the actual price change:

The convexity benefit column is always positive. Whether yields rise or fall, the actual outcome is always better than the linear estimate. When yields fall, the bond gains more than predicted. When yields rise, it loses less. For a 100 basis point drop, the bond gains €470,265, but for a 100 basis point rise, it loses only €444,172. The net advantage is €26,093.

The chart below plots the convexity benefit from the table. The U-shape shows that the benefit is negligible for small yield moves and grows rapidly as yields move further from the starting point, in either direction.

This asymmetry is a direct consequence of convexity. But it raises a question: if convexity always benefits the bondholder, why doesn’t everyone simply buy the most convex bonds available?

The answer is that convexity is not free. In an efficient market, bonds with higher convexity trade at a lower yield than bonds of the same duration but lower convexity. The bondholder pays for convexity through lower carry. Every day that rates do not move significantly, the investor in the high convexity bond earns less income than an investor holding a lower convexity bond with the same duration. The convexity benefit only materialises when rates move by enough to offset this ongoing cost in carry.

This is a no-free-lunch result. Convexity is valuable in volatile rate environments. In calm markets, it is a small drag on returns.

13.3 When convexity matters

For day-to-day portfolio management, where yield moves are typically a few basis points, the PV01 approximation is accurate enough and convexity can be safely ignored. For stress testing, scenario analysis, or markets experiencing large moves (the kind of 100 to 200 basis point shifts seen in 2022, for instance), ignoring convexity meaningfully overstates potential losses and understates potential gains.

The full price-change approximation, including the convexity correction, is:

where $C$ is the bond’s convexity. The equation above is obtained by performing a 2nd-order Taylor expansion on the relative bond price as a function of YTM. The second term is always positive, since $(\Delta y)^2$ is positive regardless of whether yields rise or fall. This confirms that convexity always adds to the bond’s price relative to the linear estimate.

13.4 The effect of coupon on duration and convexity

Both duration and convexity depend on the timing and size of a bond’s cash flows. The same logic from section 12 applies.

Higher coupon means shorter duration and lower convexity. A high coupon bond distributes its value across many near-term payments. Those short-dated cash flows are relatively insensitive to yield changes, pulling both duration and convexity down.

Lower coupon means longer duration and higher convexity. A low coupon bond concentrates its value in a single large payment at maturity. That distant cash flow is highly sensitive to yield changes and exhibits more curvature in its price-yield relationship.

These relationships have practical consequences. A pension fund matching long-dated liabilities wants high duration and may prefer zero coupon bonds because they concentrate sensitivity precisely where it is needed. A corporate treasurer hedging a near-term liability cares about PV01 and will find that lower coupon bonds deliver more PV01 per euro invested, requiring a smaller cash outlay to achieve the same hedge.

14. Key takeaways

• A bond is a lending contract in which the issuer commits to paying coupons and returning face value at maturity. Four terms define any bond: issuer, face value, coupon, and maturity date.
• Government bonds are the benchmark risk-free instruments in each currency. Corporate bonds carry a credit spread above the risk-free rate to compensate for default risk. Supranational and agency bonds sit between the two.
• Fixed rate bonds pay a constant coupon. Floating rate notes reset their coupon periodically against a benchmark rate such as EURIBOR. Zero coupon bonds pay no coupons and are purchased at a discount to face value.
• At issuance, a bond’s coupon is set at approximately the relevant swap rate plus the issuer’s credit spread, targeting a price at or near par.
• The dirty price is the full settlement amount and includes accrued interest. The clean price is the quoted market price, net of accrued interest. The two differ whenever a bond trades between coupon dates.
• In its simplest form, a bond’s fair value is the sum of its discounted cash flows: $\text{Price} = \sum C_i \times \text{DF}(t_i)$. Each cash flow uses the discount factor for its specific payment date. Fair value in financial statements can add adjustments, for example for credit risk.
• Yield to maturity (YTM) is the single constant rate that equates a bond’s price to its discounted cash flows. It is a convenient quotation mechanism and the bond’s IRR, but it assumes a flat yield curve and reinvestment of coupons at the YTM rate.
• PV01 is the change in a bond’s price for a one basis point move in yield: $\text{PV01} = D_{Mod} \times \text{Price} \times 0.0001$. It is expressed in currency terms, making it additive across a portfolio and the standard tool for measuring and aggregating interest rate risk.
• For two bonds of the same maturity, the one with the lower coupon will always have a lower PV01 in absolute terms but a higher sensitivity to yield changes relative to its price. This is because a lower coupon concentrates more of the bond’s value in the final payment, where discounting has the greatest impact.
• Macaulay duration is the weighted-average time to receive the bond’s cash flows, using present values as weights. It explains *why* sensitivity differs across bonds: the further the economic centre of gravity, the greater the impact of a yield change. For a zero coupon bond, Macaulay duration equals the time to maturity exactly.
• Modified duration converts Macaulay duration into a percentage price sensitivity: a 1% rise in yield causes the price to fall by approximately $D_{Mod}$%. It is linked to Macaulay duration by $D_{Mod} = D_{Mac} / (1 + y)$.
• Convexity captures the curvature in the price-yield relationship that duration alone misses. For a given magnitude of yield change, the price gain from a fall in yields exceeds the price loss from an equivalent rise. However, this benefit is not free: bonds with higher convexity trade at a lower yield, so the bondholder pays for convexity through lower carry every day that rates do not move significantly.