Interest-Rate Risk: Duration, Convexity, and Key-Rate Exposures

Introduction

Interest-rate risk is the possibility that changes in market yields will reduce the value, income, liquidity, or economic capital of a fixed-income position. It is central to bond portfolio management because bond prices and yields generally move in opposite directions, while the magnitude of the price response depends on maturity, coupon structure, cash-flow timing, embedded options, yield-curve shape, and the spread demanded by investors.

A single duration statistic is useful but incomplete. Macaulay and modified duration summarize sensitivity to a parallel movement in a yield measure. Effective duration is more appropriate when cash flows may change as rates change, as with callable or putable bonds. Spread duration isolates sensitivity to a change in credit or option-adjusted spread. Key-rate duration decomposes total exposure across particular maturities and is therefore better suited to nonparallel yield-curve shifts. Convexity improves the approximation for larger yield changes and explains why the price–yield relationship is curved rather than linear.

The practical objective is not merely to calculate a statistic. It is to identify the source of risk, select a measure consistent with the instrument and valuation model, interpret the result under plausible scenarios, and design a hedge whose residual exposures are understood.

Learning objectives

By the end of this chapter, the reader should be able to:

• explain the economic meaning and assumptions of Macaulay, modified, effective, spread, and key-rate duration;
• calculate duration and convexity for standard fixed-rate bonds;
• distinguish parallel from nonparallel yield-curve risk;
• assess the limitations of duration-based approximations;
• construct and evaluate duration and key-rate hedges; and
• apply governance and model-risk controls to interest-rate risk measurement.

1. The price–yield relationship

For a bond with promised cash flows Cₜ at dates t = 1, …, n, periodic yield y, and m compounding periods per year, the price is

Here, P is the clean or dirty price according to the cash-flow convention used; Cₜ is the cash flow at period t; y is the annual nominal yield consistent with the compounding frequency; and t is measured in years. The formula assumes a deterministic cash-flow schedule and a single discount rate applied to all cash flows. In a term-structure framework, each cash flow is discounted using its own spot rate rather than one yield.

The first derivative is negative: an increase in yield lowers price. The second derivative is normally positive for an option-free bond: the price decline from a yield increase is smaller in absolute value than the price increase from an equal yield decrease. This curvature is convexity.

A bond's interest-rate exposure is determined by the timing and uncertainty of its cash flows. Holding yield and credit quality constant, a long-maturity, low-coupon bond usually has greater duration than a short-maturity, high-coupon bond. A zero-coupon bond has Macaulay duration equal to its maturity because all value is received at one date.

2. Macaulay duration

Macaulay duration is the present-value-weighted average time at which a bond's cash flows are received:

where D_M is Macaulay duration in years, PV(Cₜ) is the present value of cash flow Cₜ, and P is the bond price. For periodic cash flows, t may be expressed in coupon periods, in which case the result must be divided by m to obtain years.

The weights are:

so that ∑wₜ = 1 and D_M = ∑t × wₜ. The calculation assumes fixed, known cash flows and a single yield used for discounting. It is primarily a timing measure, not a direct percentage price-sensitivity measure.

Macaulay duration has an important interpretation in immunization. If a portfolio's asset duration is matched to the duration of a liability, a small parallel yield change has approximately offsetting effects on the present values of assets and liabilities, provided the cash-flow and reinvestment assumptions are satisfied.

3. Modified duration

Modified duration converts Macaulay duration into an approximate percentage price sensitivity to a change in yield:

For a small change Δy in the annual nominal yield, with the same compounding convention,

The yield change must be expressed as a decimal: 50 basis points equals 0.0050. Modified duration is dimensionally measured in years, while Δy is dimensionless; therefore the product is a dimensionless percentage change.

The exact local relationship is

This measure is appropriate for option-free instruments whose cash flows do not change when yields move. It is less reliable for instruments with embedded options, where a yield change can alter expected exercise behavior and therefore alter cash flows.

Worked example 1: duration estimate

Consider a three-year annual-pay bond with face value 1,000, annual coupon rate 6%, and yield to maturity 5%. Its cash flows are 60, 60, and 1,060. The price is approximately:

The present values are approximately 57.14, 54.42, and 915.67. Thus:

With annual compounding, modified duration is:

For a 25-basis-point increase in yield, the estimated percentage price change is:

The estimated price decline is therefore 0.673%, or approximately 6.91 monetary units on a price of 1,027.23. The approximation ignores convexity and assumes unchanged cash flows and spread.

4. Effective duration

Effective duration measures sensitivity when cash flows may change as rates change. It is calculated by revaluing the security under an upward and downward shift in the relevant benchmark curve:

P₀ is the current price, P₋ is the price after a downward shift of Δy, and P₊ is the price after an upward shift. The shift is commonly applied to the benchmark yield curve, with the security's spread held constant, although implementation conventions differ.

Effective duration is essential for callable bonds, mortgage-backed securities, and other instruments with path-dependent or contingent cash flows. A fall in rates may increase the probability that a callable bond is redeemed, limiting its price appreciation. Consequently, its effective duration may decline and its convexity may become negative over part of the yield range.

The measure is model-dependent. It depends on the interest-rate tree or simulation model, volatility assumptions, prepayment or exercise behavior, discounting framework, and treatment of spreads. It should therefore be accompanied by scenario analysis rather than treated as an unquestionable observed fact.

5. Spread duration

Spread duration measures sensitivity to a change in a credit spread, liquidity spread, or option-adjusted spread rather than to a change in the benchmark risk-free curve. A local definition is:

where s is the spread expressed as a decimal rate. A finite-difference estimate is:

For a small spread change Δs, the approximate price effect is:

For a simple option-free credit bond, spread duration may be close to modified duration, but the two are not conceptually identical. A bond can experience a risk-free-rate shift and a spread shift simultaneously. A useful first-order decomposition is:

where D_y is duration to the benchmark curve and D_s is spread duration. The decomposition is only approximate and may omit cross-effects and convexity.

6. Convexity

Duration gives the tangent-line approximation to the price–yield curve. Convexity captures the curvature. For an option-free bond with annual compounding, modified convexity can be expressed as:

C is measured in years squared when y is a decimal annual yield and t is in annual periods. For m compounding periods, the equivalent expression must consistently incorporate m and the period count; mixing annual and periodic units is an error.

The second-order approximation is:

The first term is linear and the second is always positive for positive convexity. The approximation is more accurate for small yield changes and for instruments whose cash flows remain fixed.

Convexity has several interpretations. It reduces the actual loss from a yield increase relative to the duration-only estimate and increases the actual gain from a yield decrease. All else equal, investors generally value positive convexity. However, positive convexity is not free: it is commonly associated with lower yield or higher price. Callable bonds can exhibit negative convexity because the issuer's call option limits upside when rates fall.

For a portfolio, value-weighted convexity is appropriate when exposures are expressed consistently:

where Vᵢ is the market value of position i and V_P is total portfolio value. Portfolio convexity also includes interaction effects when instruments are revalued under a common curve model.

7. From parallel shifts to key-rate exposures

Traditional duration assumes that all relevant yields move by the same amount. Actual yield curves often twist, steepen, flatten, or develop localized movements. A portfolio can therefore have near-zero total duration while remaining highly exposed to a 2-year or 30-year rate movement.

Key-rate duration (KRD) measures sensitivity to a shift at one maturity node while other nodes are held fixed or interpolated according to a specified curve methodology. A finite-difference definition is:

where k identifies the key maturity, P_k⁻ is the portfolio value after a downward shift at node k, P_k⁺ is the value after an upward shift, and Δy_k is the node shift. KRD is usually reported in years.

For a vector of small node changes Δy_k, the first-order portfolio return is:

The sum of key-rate durations is approximately, but not necessarily exactly, the portfolio's parallel-shift duration. Differences arise from interpolation, curve construction, instrument conventions, and whether the key-rate shocks reproduce a uniform shift.

KRD is a local risk measure. It depends on selected nodes, interpolation rules, discounting and projection curves, and the treatment of cash flows between nodes. A detailed KRD report should state these conventions.

Key-rate hedging

Suppose a portfolio has key-rate vector K and a hedge instrument has key-rate vector h. If x units of one hedge are used, the residual exposure is K − xh. With multiple hedges, the objective may be to solve:

where H is a matrix whose columns contain the hedge instruments' key-rate durations and x is the vector of hedge notionals. Exact matching may be impossible, so a least-squares or constrained optimization approach is often used. A practical objective minimizes expected residual variance while considering hedge cost, liquidity, transaction limits, basis risk, and capital usage.

8. Worked example 2: duration hedge

Assume a bond portfolio has market value 50 million and modified duration 5.2. A futures contract used as a hedge has an underlying deliverable basket with market value exposure of 100,000 per contract and modified duration 7.0. The portfolio dollar duration is:

The approximate dollar duration per futures contract is:

The number of contracts required to offset a parallel rise in rates is:

A short position of approximately 371 contracts is an initial hedge. The exact trade must account for contract conversion factors, futures price, delivery option, margining, accrued interest, and the basis between the portfolio curve and the futures basket. Rounding to 371 contracts leaves residual risk; a risk manager should report that residual rather than claiming a perfect hedge.

9. A practical risk-measurement framework

A robust process contains five stages.

First, define the valuation basis. Specify clean or dirty price, settlement date, day-count convention, benchmark curve, spread convention, compounding frequency, and currency.

Second, classify cash flows. Determine whether cash flows are fixed, floating, callable, putable, prepayable, default-sensitive, or otherwise contingent.

Third, select the risk measures. Use modified duration and analytical convexity for simple option-free bonds; effective duration and effective convexity for instruments with material optionality; spread duration for spread shocks; and KRD for curve-shape risk.

Fourth, stress the portfolio. Include parallel shifts, steepening and flattening, twists, historical shocks, and severe but plausible scenarios. Nonlinear instruments require full revaluation or a sufficiently rich approximation.

Fifth, hedge and monitor. Hedge the exposures that matter economically, not merely the headline duration. Rebalance after market movements, portfolio trades, roll-down, coupon payments, option exercise, and changes in model assumptions.

A useful risk report includes market value, duration, dollar duration, convexity, spread duration, KRD by node, scenario P&L, hedge positions, liquidity, and model version. Limits should be stated in both aggregate and bucketed terms.

10. Illustrative case: a liability-driven investor

Background

A pension fund owes a large payment in approximately 20 years. Its assets include corporate bonds concentrated around 5–10 years and a smaller allocation to long government bonds. The portfolio's aggregate modified duration is 11 years, while the liability duration is 17 years.

Problem

A parallel rise in yields is expected to reduce both asset and liability values, but the longer-duration liability will decline more. The fund's economic surplus is therefore vulnerable to rising rates. In addition, the asset portfolio has a pronounced 7-year KRD and insufficient 20-year KRD.

Analysis

The fund first calculates dollar duration for assets and liabilities. It then decomposes both into 2-, 5-, 10-, 20-, and 30-year KRD buckets. A parallel duration match alone would conceal the mismatch: the asset and liability totals could be similar after a trade while their bucket exposures remain different.

Approach

The fund purchases long-duration government bonds and receives fixed in interest-rate swaps at selected maturities, while reducing part of its intermediate corporate-bond exposure. The hedge is calibrated to reduce the 10- and 20-year KRD mismatch, not simply to set total duration equal to the liability duration. Spread duration is monitored separately because swaps hedge benchmark rates, not corporate credit spreads.

Outcome

Under a parallel shift, the surplus becomes less sensitive. Under a 7s/20s curve twist, the KRD hedge performs substantially better than the original aggregate-duration hedge. However, the fund retains basis risk, swap collateral requirements, corporate spread exposure, and potential liquidity risk during stressed markets.

Learning points

The case demonstrates that duration matching is necessary but not sufficient. The relevant hedge depends on the liability structure, the yield curve, the instrument used, and the distinction between risk-free-rate and spread exposure.

11. Limitations, governance, and contemporary practice

Duration is local. A large rate move can make a first-order estimate materially inaccurate. Convexity improves the estimate but may itself change as rates move. For mortgage-backed securities and callable bonds, effective duration can change rapidly because exercise and prepayment behavior changes.

Model risk is significant. Curve construction, interpolation, volatility, credit migration, default assumptions, prepayment models, and liquidity adjustments can all affect reported risk. Independent validation should test formulas, market-data inputs, scenario construction, and stability under alternative assumptions. Back-testing compares predicted and realized sensitivities, although realized P&L also contains spread, carry, roll-down, and trading effects.

A hedge can create new risks. Futures introduce basis and margin risk; swaps introduce counterparty, collateral, funding, and benchmark-transition risk; bonds introduce liquidity and credit risk. Governance should therefore include approved instruments, counterparty limits, collateral terms, stress limits, documentation, and escalation procedures.

Technology enables automated revaluation, scenario grids, factor models, and real-time KRD dashboards. Machine-learning tools may assist with anomaly detection or forecasting, but they do not eliminate the need for economically interpretable measures, stable data lineage, human oversight, and independent model validation. AI-generated recommendations should not be used as autonomous hedging instructions without controls, auditability, and suitability review.

Best practice is to combine analytical sensitivities with full revaluation, especially for nonlinear instruments. Reports should distinguish market-rate P&L from carry, roll-down, spread movement, foreign-exchange effects, and trading activity. Risk measures should be recalculated after material portfolio changes and reviewed under stressed liquidity conditions.

Chapter Summary

Interest-rate risk arises because the present value of fixed-income cash flows changes when discount rates, benchmark yields, or spreads change. Macaulay duration is the present-value-weighted timing of cash flows. Modified duration converts that timing measure into an approximate percentage price sensitivity for fixed cash flows. Effective duration is preferred when rates can change the expected cash flows of an instrument. Spread duration isolates sensitivity to credit or other spreads.

Duration provides a first-order, generally linear approximation. Convexity captures the second-order curvature of the price–yield relationship and is especially important for larger shocks and option-sensitive instruments. Aggregate duration assumes a parallel shift, whereas key-rate duration identifies exposure to movements at individual curve maturities. Key-rate analysis is therefore essential for managing steepening, flattening, and localized curve movements.

Professional risk management combines clearly defined valuation conventions, appropriate sensitivity measures, scenario analysis, hedge optimization, model validation, and governance. A successful hedge reduces the intended exposure without creating unacceptable basis, liquidity, counterparty, collateral, or model risks.

Key Terms

Macaulay duration: Present-value-weighted average time to receive a bond's cash flows.

Modified duration: Approximate percentage price sensitivity to a change in yield for fixed cash flows.

Effective duration: Duration estimated by revaluing an instrument when rates change and cash flows may change.

Spread duration: Sensitivity of price to a change in a credit, liquidity, or option-adjusted spread.

Convexity: Second-order curvature of the price–yield relationship.

Dollar duration: Market-value change associated with a one-unit change in yield, approximately equal to market value multiplied by duration.

Key-rate duration: Sensitivity to a yield change at a specified maturity node.

Parallel shift: Equal change in yields at all maturities.

Steepening: A widening difference between long- and short-maturity yields.

Flattening: A narrowing difference between long- and short-maturity yields.

Basis risk: Risk that a hedge and the exposure do not respond identically.

Yield to maturity: Single discount rate that equates a bond's price with its promised cash flows under stated conventions.

Spot rate: Discount rate applicable to a cash flow at a particular maturity.

Immunization: Strategy intended to protect an asset–liability position from specified interest-rate changes.

Embedded option: Contractual feature allowing an issuer or investor to alter cash flows under stated conditions.

Option-adjusted spread: Spread measure that attempts to remove the value of an embedded option.

Full revaluation: Repricing an instrument under a new market scenario rather than applying a local approximation.

Dollar duration hedge: Hedge designed to offset the value-weighted duration exposure of a position.

Review Questions

Short Answer Questions

1. What economic relationship does modified duration summarize?

1. Why is effective duration preferable to modified duration for a callable bond?

1. Define spread duration and distinguish it from benchmark-yield duration.

1. What additional information does convexity provide beyond duration?

1. Why can a portfolio with low aggregate duration still have substantial interest-rate risk?

Long Answer Questions

1. Derive the relationship between Macaulay duration and modified duration, and explain the assumptions required for the relationship to be useful.

1. Compare analytical duration, effective duration, and key-rate duration in terms of inputs, assumptions, and appropriate applications.

1. Explain how positive and negative convexity affect the performance of option-free and callable bonds.

1. Describe a complete governance framework for measuring and hedging interest-rate risk in a large fixed-income portfolio.

1. Explain why a duration-matched hedge may fail under a nonparallel yield-curve movement.

Analytical / Application Questions

1. A bond portfolio worth 80 million has modified duration 6.0. Estimate the percentage and monetary price change from a 40-basis-point yield increase, ignoring convexity. State the limitation of the estimate.

1. A portfolio has KRDs of 1.0, 2.5, and 4.0 years at the 2-, 10-, and 30-year nodes. The corresponding yield changes are +20, +10, and −15 basis points. Estimate the percentage price change using the first-order KRD approximation.

1. Design a hedge for a liability portfolio whose aggregate duration is matched but whose 10-year KRD is too high and 30-year KRD is too low. Explain why a single short-term futures hedge may not solve the problem and identify additional information required before trading.

Multiple Choice Questions with Answers

Question 1

For an option-free bond with fixed cash flows, modified duration primarily measures:

A. The bond’s default probability

B. Approximate percentage price sensitivity to a yield change

C. The average credit spread across issuers

D. The probability of early redemption

Correct Answer: B. Approximate percentage price sensitivity to a yield change

Question 2

A 50-basis-point yield increase is represented in decimal form as:

A. 0.0005

B. 0.0050

C. 0.0500

D. 0.5000

Correct Answer: B. 0.0050

Question 3

Which duration measure is most appropriate when a rate change can alter expected cash flows?

A. Macaulay duration

B. Modified duration

C. Effective duration

D. Settlement duration

Correct Answer: C. Effective duration

Question 4

If a bond has modified duration of 4.0 and its yield rises by 25 basis points, the duration-only estimated price change is:

A. +1.00%

B. +0.10%

C. −0.10%

D. −1.00%

Correct Answer: D. −1.00%

Question 5

Convexity is included in a price-change approximation primarily because:

A. Bond prices are linear in yields

B. The price–yield relationship is curved

C. Credit spreads are always zero

D. Cash flows are always uncertain

Correct Answer: B. The price–yield relationship is curved

Question 6

Spread duration measures sensitivity to:

A. A change in the bond’s coupon rate

B. A change in a benchmark yield only

C. A change in a credit or related spread

D. A change in face value

Correct Answer: C. A change in a credit or related spread

Question 7

Key-rate duration is especially useful for analyzing:

A. Only default losses

B. Nonparallel yield-curve movements

C. Coupon payment dates only

D. Foreign-exchange translation

Correct Answer: B. Nonparallel yield-curve movements

Question 8

A zero-coupon bond’s Macaulay duration, under standard assumptions, is generally equal to its:

A. Coupon rate

B. Yield spread

C. Maturity

D. Convexity

Correct Answer: C. Maturity

Question 9

A short position in a duration-sensitive futures contract is generally used to hedge a bond portfolio against:

A. A rise in interest rates

B. A fall in credit quality only

C. A decline in inflation only

D. An increase in coupon income

Correct Answer: A. A rise in interest rates

Question 10

A portfolio has modified duration 5 and market value 10 million. Ignoring convexity, a 10-basis-point yield increase produces an estimated value change of:

A. −5,000

B. −50,000

C. −500,000

D. +50,000

Correct Answer: B. −50,000