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Duration and convexity are the two fundamental measures of a bond's sensitivity to changes in interest rates. Together they allow investors to estimate how much a bond's price will change when yields move, enabling precise interest rate risk management in fixed income portfolios. Duration has two primary forms. Macaulay Duration, introduced by Frederick Macaulay in 1938, is the weighted average time to receive the bond's cash flows (coupon payments and principal), with weights proportional to the present value of each cash flow. It is expressed in years and represents the effective 'time to payback' of the bond's investment. A zero-coupon bond's Macaulay Duration equals its maturity since all cash flow is received at the end; a coupon bond's Macaulay Duration is shorter than maturity because interim coupon payments reduce the effective waiting time. Modified Duration converts Macaulay Duration into a direct measure of price sensitivity: it estimates the percentage change in bond price for a 1% (100 basis point) change in yield. Modified Duration = Macaulay Duration / (1 + y/m), where y is the yield to maturity and m is the number of coupon periods per year. For example, a bond with Modified Duration of 7 is expected to fall approximately 7% in price if yields rise 100 basis points — or rise 7% if yields fall 100 basis points. Convexity is the second-order measure of price sensitivity — it captures the curvature in the price-yield relationship that duration alone misses. Because bond prices and yields have a convex (curved) relationship rather than a linear one, duration underestimates price gains when yields fall and overestimates price declines when yields rise. Adding the convexity adjustment produces a more accurate price change estimate, especially for large yield movements. Positive convexity (characteristic of standard bonds) means the bond benefits from interest rate movements in both directions relative to the linear duration approximation. Mortgage-backed securities can exhibit negative convexity when falling rates trigger prepayments. These concepts are foundational to bond portfolio immunization, liability-driven investment (LDI) strategies, and fixed income risk management at banks, insurance companies, and pension funds.
Modified Duration = Macaulay Duration / (1 + y/m) ΔP/P ≈ −D_mod × Δy + 0.5 × C × (Δy)². This formula calculates duration convexity by relating the input variables through their mathematical relationship. Each component represents a measurable quantity that can be independently verified.
- 1List all of the bond's future cash flows with their timing: coupon payments (Coupon = Face Value × Coupon Rate / m) at each period and the face value repayment at maturity.
- 2Discount each cash flow to its present value using the yield to maturity: PV_t = CF_t / (1 + y/m)^t, where t is the period number.
- 3Sum all present values to obtain the bond's price: P = Σ PV_t.
- 4Compute Macaulay Duration: D_Mac = Σ [t × PV_t] / P — the weighted average time (in periods), then divide by m to convert to years.
- 5Compute Modified Duration: D_mod = D_Mac / (1 + y/m). This is the bond's first-order price sensitivity — multiply by −1 and the yield change to estimate the percentage price change.
- 6Compute Convexity: C = [Σ t(t+1) × PV_t / (1+y/m)²] / (P × m²). This second-order term adjusts the duration estimate for the curvature of the price-yield curve.
- 7For a given yield change Δy: estimate ΔP/P ≈ −D_mod × Δy + 0.5 × C × (Δy)². For small Δy (< 25 bps), the duration term alone is usually sufficient; for larger changes, convexity becomes material.
At-par bond: price = $1,000. A 100bp yield rise → approx. -7.93% price change.
When a bond's coupon rate equals its yield to maturity, it trades at par ($1,000). The Macaulay Duration of 8.11 years reflects the weighted average time to receive cash flows across 10 years of semi-annual coupons plus final principal. Modified Duration of 7.93 means the bond loses approximately 7.93% in price for each 1% rise in yields. For a 100 basis point yield increase: ΔP/P ≈ −7.93% × 1% + 0.5 × 79.5 × (0.01)² ≈ −7.93% + 0.40% = −7.53%, showing how convexity partially offsets the duration decline.
Zero-coupon: Duration equals maturity. Highest duration for a given maturity — maximum rate sensitivity.
A zero-coupon bond pays no interim coupons — all cash flow arrives at maturity. Therefore, its Macaulay Duration exactly equals its maturity (20 years). With a Modified Duration of 19.05, a 1% yield increase causes approximately a 19.05% price decline — making this bond extremely sensitive to interest rate changes. The very high convexity of 399.7 provides substantial protection against large yield moves, but the bond remains the most interest-rate-sensitive instrument for its maturity. Zero-coupon bonds are used extensively in liability-driven investment strategies to match long-dated liabilities precisely.
High coupon relative to yield — trades above par (premium bond). Low rate sensitivity.
With a 6% coupon and 5% YTM, this bond trades above par (premium bond, price ≈ $1,027.23). The high coupon accelerates cash flow receipt, reducing Macaulay Duration to 2.74 years despite a 3-year maturity. Modified Duration of 2.67 means only a 2.67% price change per 1% yield move — making this bond far less interest rate sensitive than the 10-year Treasury or zero-coupon bond. Short-duration corporate bonds are used to reduce interest rate risk while still earning credit spreads above government bond yields.
Portfolio duration is the value-weighted average of component durations.
Portfolio duration = 0.40 × 4.3 + 0.60 × 11.2 = 1.72 + 6.72 = 8.44. This bond ladder combining short and long bonds achieves an intermediate duration of 8.44 years — similar to a 10-year Treasury — without holding a single 10-year bond. Laddering provides liquidity benefits (bonds mature periodically) and allows precise duration targeting. A portfolio manager targeting a duration of 8.0 could adjust the weights between Bond A and Bond B to achieve the exact target using simple linear algebra.
Bond portfolio immunization: matching asset duration to liability duration for pension funds and insurance companies. This application is commonly used by professionals who need precise quantitative analysis to support decision-making, budgeting, and strategic planning in their respective fields
Interest rate hedging: calculating Treasury futures or interest rate swap hedge ratios using DV01. Industry practitioners rely on this calculation to benchmark performance, compare alternatives, and ensure compliance with established standards and regulatory requirements
Fixed income risk management: computing portfolio duration and DV01 limits for trading desks. Academic researchers and students use this computation to validate theoretical models, complete coursework assignments, and develop deeper understanding of the underlying mathematical principles
Liability-Driven Investment (LDI) strategy construction for defined benefit pension plan assets. Financial analysts and planners incorporate this calculation into their workflow to produce accurate forecasts, evaluate risk scenarios, and present data-driven recommendations to stakeholders
Regulatory capital: duration-based interest rate risk capital calculations under Basel III Pillar 1 and IRRBB standards. This application is commonly used by professionals who need precise quantitative analysis to support decision-making, budgeting, and strategic planning in their respective fields
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in bond duration & convexity calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in bond duration & convexity calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
In practice, this edge case requires careful consideration because standard assumptions may not hold. When encountering this scenario in bond duration & convexity calculator calculations, practitioners should verify boundary conditions, check for division-by-zero risks, and consider whether the model's assumptions remain valid under these extreme conditions.
| Bond Type | Maturity | Modified Duration | Convexity | Price Sensitivity to +100bps |
|---|---|---|---|---|
| 3-Month T-Bill | 3 months | 0.25 | 0.06 | −0.25% |
| 2-Year Treasury Note | 2 years | 1.85 | 3.9 | −1.85% |
| 5-Year Treasury Note | 5 years | 4.40 | 21.0 | −4.40% |
| 10-Year Treasury Bond | 10 years | 7.95 | 79.5 | −7.55% |
| 30-Year Treasury Bond | 30 years | 16.50 | 367.0 | −14.66% |
| 10-Year Zero-Coupon Bond | 10 years | 9.52 | 98.0 | −9.05% |
| IG Corporate Bond (10yr) | 10 years | 7.40 | 72.0 | −7.04% |
| High-Yield Bond (5yr) | 5 years | 3.80 | 16.5 | −3.61% |
| MBS Pass-Through | ~30yr nominal | 3.0 – 5.0 | Negative | −3% to −5% |
What is the practical difference between Macaulay and Modified Duration?
Macaulay Duration is expressed in years and represents the weighted average time to receive the bond's cash flows — it is a measure of time, not directly a price sensitivity measure. Modified Duration is the first derivative of the bond price with respect to yield change (divided by price), and directly measures the percentage price change for a 1% change in yield. For practical applications — estimating how much a bond's value will change when rates move — Modified Duration is the relevant measure. Macaulay Duration is more useful conceptually (understanding the timing of cash flows) and for constructing immunized portfolios (matching asset duration to liability duration in years).
Why does convexity matter and when does it become important?
Convexity captures the curvature in the price-yield relationship. Duration alone provides a linear approximation of price changes, but the actual price-yield curve is convex — it curves upward. For small yield changes (less than 25 basis points), duration is usually sufficient. For larger yield movements (50–200+ basis points), convexity becomes increasingly material. During the dramatic rate movements of 2022 (the Federal Reserve raised rates by 425 basis points in one year), investors with properly calculated convexity were far more accurate in estimating portfolio losses than those relying on duration alone. Higher convexity is generally desirable as it means the bond loses less when rates rise and gains more when rates fall.
What is negative convexity and which bonds exhibit it?
Most standard bonds have positive convexity — price gains accelerate when yields fall and price losses slow when yields rise. Negative convexity is the reverse: price gains slow when yields fall and price losses accelerate when yields rise. This occurs with callable bonds (the issuer calls the bond when rates fall, capping price appreciation) and mortgage-backed securities (homeowners prepay mortgages when rates fall, shortening the security's duration just when you want it long). Inverse floaters can also exhibit negative convexity. Securities with negative convexity typically offer higher yields to compensate investors for this unfavorable characteristic.
How is duration used in bond portfolio immunization?
Immunization is the strategy of matching a portfolio's duration to the duration of its target liabilities, so that changes in interest rates affect both assets and liabilities equally — leaving the funding surplus unchanged. A pension fund with liabilities having an effective duration of 12 years would target a bond portfolio with Modified Duration of 12 years. When rates rise, both asset values and liability present values decline proportionally, leaving the funding ratio approximately unchanged. Full immunization also requires matching convexity: if the asset convexity exceeds liability convexity, the portfolio is positively positioned (gains from any yield change). Liability-Driven Investment (LDI) strategies used by pension funds and insurance companies are built on this framework.
What is Dollar Duration (DV01) and how is it used?
Dollar Duration, commonly expressed as DV01 (Dollar Value of a Basis Point) or PVBP (Price Value of a Basis Point), measures the change in a bond's price in dollar terms for a 1 basis point (0.01%) change in yield. DV01 = Modified Duration × Bond Price × 0.0001. For example, a bond with Modified Duration of 8 and price of $1,000,000 has DV01 = 8 × $1,000,000 × 0.0001 = $800. DV01 is used extensively by fixed income traders and portfolio managers to size positions, calculate hedge ratios, and communicate risk in dollar terms rather than percentages — which is more intuitive for large portfolios and trading desks.
How does coupon rate affect duration?
Higher coupon rates produce shorter durations, and lower (or zero) coupon rates produce longer durations, for bonds with the same maturity. This is because higher coupons mean the investor receives a greater proportion of total cash flows earlier (as interim coupon payments), reducing the weighted average time to cash flow receipt (Macaulay Duration). A high-coupon bond is therefore less interest-rate sensitive than a low-coupon bond of the same maturity. This relationship explains why zero-coupon bonds have the maximum possible duration (equal to maturity) for any given maturity length and yield level.
How does yield level affect duration and convexity?
Duration and convexity are both sensitive to the prevailing yield level. When yields are low, duration is generally higher (because discount rates are low, making distant cash flows more valuable), and convexity is also higher. When yields are high, duration is lower and convexity is lower. This is particularly relevant in the current environment where rates have risen significantly from near-zero levels — portfolios that appeared to have high duration risk in 2020–2021 now have lower effective duration at higher yield levels. This yield-dependency means duration must be recalculated whenever yields move significantly, and static duration estimates can become stale quickly in volatile rate environments.
プロのヒント
For practical fixed income risk management, use Dollar Duration (DV01) to communicate risk in monetary terms rather than percentage terms. A portfolio with $5,000 DV01 changes in value by $5,000 for every basis point move in yields — immediately interpretable by traders, risk managers, and portfolio managers.
ご存知でしたか？
Frederick Macaulay's 1938 paper introducing Macaulay Duration was part of a landmark study commissioned by the National Bureau of Economic Research on U.S. interest rates spanning more than a century. The duration concept was largely ignored for nearly three decades until immunization theory revived interest in it during the 1960s and 1970s.