Duration & Price Sensitivity — Duration concept and price sensitivity (intro) - Finance

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Historical Context & Motivation

Fixed-income markets have long confronted a fundamental challenge: how to quantify the risk that rising or falling interest rates pose to the value of a bond portfolio. Before formal analytical tools existed, portfolio managers relied largely on intuition and maturity as a crude proxy for interest rate exposure. A bond maturing in 30 years was understood to be "riskier" than one maturing in 2 years, but the precise magnitude of price change for a given shift in yields remained elusive. The development of duration as a concept addressed this gap directly, providing a single metric that captures the weighted-average timing of a bond's cash flows and, by extension, its sensitivity to interest rate movements.

The intellectual roots of duration trace back to the 1930s, a period when economists were grappling with the Great Depression and the mechanics of fixed-income valuation were receiving renewed scrutiny. Over the subsequent decades, the concept was refined, extended, and eventually became a cornerstone of modern portfolio management and risk measurement. Understanding this history illuminates why duration remains one of the most widely cited statistics in bond analysis today.

1938 • Macaulay Introduces Duration

Frederick Macaulay publishes his seminal study on bond yields, introducing Macaulay duration as the weighted-average time to receive a bond's cash flows, measured in years. This gave analysts the first rigorous tool for comparing interest rate sensitivity across bonds.

1952 • Redington & Immunization

Frank Redington, a British actuary, develops the concept of immunization — matching the duration of assets and liabilities to protect a portfolio against interest rate shifts. This marked the first practical application of duration in institutional portfolio management.

1966 • Modified Duration Formalized

Financial economists formally derive modified duration by dividing Macaulay duration by (1 + y/k), linking the time-weighted metric directly to the percentage price change of a bond for a small change in yield.

1972 • Duration Enters Trading Desks

As interest rate volatility increased in the 1970s, bond traders adopted duration-based hedging strategies. The concept migrated from academic journals to Wall Street, becoming central to fixed-income risk management.

1983 • Convexity Extends the Framework

Researchers introduce convexity as a second-order correction to duration, capturing the curvature in the price–yield relationship. Together, duration and convexity provide a much more accurate estimate of bond price changes for larger yield shifts.

The central question that duration answers is deceptively simple: if interest rates change by a small amount, how much will the price of my bond change? Maturity alone cannot answer this because two bonds with identical maturities but different coupon rates or cash flow structures will respond differently to the same yield shift. Duration resolves this ambiguity by weighting each cash flow by both its present value and its timing, producing a single number that encapsulates interest rate exposure.

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Core Principles & Definitions

At its essence, duration quantifies the effective time horizon over which a bondholder receives value. It draws on the principle that a dollar received sooner is worth more than a dollar received later, and it uses present-value weighting to collapse the entire stream of coupon payments and the final principal repayment into a single summary statistic. Before exploring the mathematics in detail, it is important to internalize several foundational ideas that underpin the concept.

Macaulay Duration

The weighted-average time (in years) until a bond's cash flows are received, where each cash flow is weighted by its present value as a fraction of the bond's total price. A zero-coupon bond's Macaulay duration equals its maturity.

Modified Duration

A direct measure of price sensitivity: it tells you the approximate percentage change in a bond's price for a 1% (100 basis point) change in yield. It is derived from Macaulay duration by dividing by (1 + y/k).

Inverse Price–Yield Relationship

Bond prices and yields move in opposite directions. When market interest rates rise, the present value of future cash flows falls, reducing the bond's price. Duration quantifies the magnitude of this inverse relationship.

Coupon Effect

Higher-coupon bonds have shorter duration than lower-coupon bonds of the same maturity, because a larger share of total present value is received earlier. Zero-coupon bonds always have the longest duration for a given maturity.

Linear Approximation

Duration provides a first-order (linear) estimate of price changes. For small yield changes, it is highly accurate. For larger shifts, the curvature of the price–yield relationship (convexity) introduces approximation error that duration alone does not capture.

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KEY TAKEAWAY

Think of duration like the balance point of a seesaw loaded with weights at different positions. Each coupon payment is a weight placed at its payment date, and the principal repayment is a heavy weight at the far end. Duration tells you where the fulcrum must be placed so the seesaw balances perfectly. A bond with large, frequent coupons places more weight near the front, pulling the balance point (duration) closer to today. A zero-coupon bond concentrates all weight at maturity, so the fulcrum sits right at the end.

These five principles interact to determine every bond's interest rate risk profile. A portfolio manager who understands duration can compare bonds of different maturities, coupon structures, and credit qualities on a common basis, assess the overall sensitivity of a portfolio, and construct hedges that neutralize unwanted interest rate exposure.

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Visual Explanation — The Price–Yield Curve

The relationship between a bond's price and its yield to maturity is not linear — it follows a convex curve. Duration captures the slope of the tangent line to this curve at the current yield, which is why it serves as a linear approximation of price changes. The following diagram illustrates this relationship for a representative coupon bond.

The actual price curve is convex — it bows upward. The tangent line at the current yield represents the duration-based estimate. For small yield changes, the tangent closely tracks the curve. For larger moves, the gap (convexity effect) grows, and duration alone underestimates the true price for both rate increases and decreases.

Several insights emerge from this diagram. First, the slope of the tangent line is steeper at lower yields, meaning bond prices are more sensitive to yield changes when rates are low. Second, the convexity of the actual curve means that duration systematically overpredicts price declines (when yields rise) and underpredicts price gains (when yields fall). Third, the accuracy of the duration approximation deteriorates as the magnitude of the yield change increases, which motivates the introduction of convexity as a complementary risk measure.

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Mathematical Framework

The mathematical derivation of duration begins with the fundamental bond pricing equation and proceeds through differentiation with respect to yield. Understanding this derivation clarifies why duration works as a price sensitivity measure and highlights its inherent limitations as a linear approximation.

Bond Pricing Foundation

BOND PRICE

P = Σ [CFₜ / (1 + y/k)^(t)] for t = 1 to N

where P = bond price, CFₜ = cash flow at period t, y = annual yield to maturity, k = number of coupon payments per year, and N = total number of periods.

Macaulay Duration

MACAULAY DURATION

D_Mac = (1/P) × Σ [t × CFₜ / (1 + y/k)^t] for t = 1 to N

Each cash flow's present value is multiplied by its period number t, then summed and divided by the bond's price P. The result is expressed in periods; divide by k to convert to years.

Modified Duration

MODIFIED DURATION

D_Mod = D_Mac / (1 + y/k)

Modified duration adjusts Macaulay duration to directly estimate the percentage price change. D_Mod represents the approximate percentage decrease in price for a 1 percentage point increase in yield.

Price Sensitivity Approximation

PRICE CHANGE ESTIMATE

ΔP/P ≈ −D_Mod × Δy

where ΔP/P is the percentage price change and Δy is the change in yield (in decimal form). The negative sign reflects the inverse price–yield relationship. For example, if D_Mod = 7.5 and yields rise by 0.01 (100 bps), the estimated price decline is approximately 7.5%.

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Dollar Duration

In practice, portfolio managers also use dollar duration (or DV01, the "dollar value of one basis point"), which translates the percentage sensitivity into a dollar amount: Dollar Duration = D_Mod × P. This is especially useful for hedging, where you need to know the actual dollar exposure of a position to a given yield change.

SECTION 5

Determinants of Duration

A bond's duration is not fixed in isolation — it is shaped by the interplay of several characteristics. Understanding these determinants allows analysts to predict, without calculation, whether one bond will be more or less sensitive to rate changes than another. The three primary drivers are maturity, coupon rate, and yield to maturity.

Duration increases with maturity for all bonds, but the rate of increase depends on the coupon. The zero-coupon bond has a duration equal to its maturity (the 45° line). Coupon-paying bonds have durations that rise with maturity but at a decreasing rate, eventually approaching an asymptotic ceiling. Higher coupons (8% coupon) produce substantially shorter durations than lower coupons (4% coupon) at the same maturity.

Factor	Change	Effect on Duration	Intuition
Maturity	↑ Increase	↑ Increases	Cash flows are pushed further into the future, increasing the weighted-average time.
Coupon rate	↑ Increase	↓ Decreases	Larger coupons shift more present value to earlier periods, pulling the center of gravity forward.
Yield to maturity	↑ Increase	↓ Decreases	Higher discount rates reduce the present value of distant cash flows more than near-term ones, shortening the weighted average.
Coupon frequency	↑ More frequent	↓ Slightly decreases	More frequent payments deliver value sooner, marginally reducing duration.

Summary of factors affecting duration

An important corollary is that for premium bonds (coupon rate > yield), duration is shorter than for discount bonds of the same maturity, because the higher coupon payments front-load more present value. Conversely, deep-discount and zero-coupon bonds exhibit the greatest duration and, therefore, the highest price sensitivity to interest rate changes for their maturity class.

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Worked Example — Computing Duration & Estimating Price Change

Consider a 3-year bond with a face value of $1,000, an annual coupon rate of 6% (paid annually), and a yield to maturity of 5%. We will compute the Macaulay duration, the modified duration, and then estimate the price change if yields rise by 50 basis points.

3-Year, 6% Annual Coupon Bond at 5% YTM

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Step 1 — Identify Cash Flows

The bond pays annual coupons of $60 (6% × $1,000) for three years, plus the $1,000 par value at maturity. The cash flows are: Year 1 = $60, Year 2 = $60, Year 3 = $1,060.
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Step 2 — Compute Present Values of Each Cash Flow

Using y = 5% (0.05), we discount each cash flow: PV₁ = $60 / (1.05)¹ = $60 / 1.05 = $57.14 PV₂ = $60 / (1.05)² = $60 / 1.1025 = $54.42 PV₃ = $1,060 / (1.05)³ = $1,060 / 1.1576 = $915.75

Bond Price P = $57.14 + $54.42 + $915.75 = $1,027.31
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Step 3 — Compute Weighted Time for Each Cash Flow

Multiply each PV by its period number: t × PV₁ = 1 × $57.14 = $57.14 t × PV₂ = 2 × $54.42 = $108.84 t × PV₃ = 3 × $915.75 = $2,747.25

Sum of weighted PVs = $57.14 + $108.84 + $2,747.25 = $2,913.23
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Step 4 — Macaulay Duration

Divide the sum of weighted present values by the bond price: D_Mac = $2,913.23 / $1,027.31 = 2.836 years

Macaulay Duration = 2.836 years
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Step 5 — Modified Duration

Since payments are annual (k = 1): D_Mod = D_Mac / (1 + y/k) = 2.836 / (1 + 0.05/1) = 2.836 / 1.05 = 2.701

Modified Duration = 2.701
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Step 6 — Estimate Price Change for +50 bps

If yields rise from 5% to 5.50% (Δy = +0.005): ΔP/P ≈ −D_Mod × Δy = −2.701 × 0.005 = −0.01351 = −1.351% ΔP ≈ −0.01351 × $1,027.31 = −$13.88

Estimated new price ≈ $1,027.31 − $13.88 = $1,013.43 (approximately a 1.35% decline)

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Verification

If we reprice the bond at 5.50% YTM directly: P = 60/1.055 + 60/1.055² + 1060/1.055³ = $56.87 + $53.91 + $902.81 = $1,013.59. The duration-based estimate of $1,013.43 is remarkably close — off by only $0.16, or about 0.02%, demonstrating the accuracy of the linear approximation for small yield changes.

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Strengths & Limitations of Duration

Duration is one of the most widely used metrics in fixed-income analysis, but like any model, it comes with assumptions and boundary conditions. A sophisticated practitioner understands both what duration captures well and where it falls short.

Strengths	Limitations
Provides a single, intuitive number that summarizes interest rate risk, enabling quick comparison across bonds.	Assumes a parallel shift in the yield curve — all maturities move by the same amount, which rarely happens in practice.
Highly accurate for small yield changes (±25–50 bps), which covers most day-to-day market movements.	As a linear approximation, it becomes less accurate for large yield shifts (±200+ bps); convexity correction is needed.
Additive at the portfolio level: portfolio duration is the weighted average of individual bond durations.	Does not account for embedded options (e.g., callable or putable bonds) that alter cash flow timing. Effective duration is needed for such instruments.
Directly supports immunization strategies for liability-driven investing (matching asset and liability durations).	Ignores credit risk, liquidity risk, and other non-interest-rate factors that influence bond prices.
Easily computed and widely reported by data providers, making it accessible to all market participants.	Macaulay duration is denominated in years but does not represent a specific payment date — it is a weighted average that can be misinterpreted.

Balancing the utility and constraints of duration as a risk metric

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KEY TAKEAWAY

Think of duration as a first-generation GPS: it gives you an excellent estimate of the direction and distance to your destination (price change), but it assumes you are traveling in a straight line. For short trips (small yield changes), this is nearly perfect. For longer journeys (large yield shifts), you need to account for the curvature of the road — that is the role of convexity, the second-order adjustment that captures the nonlinearity of the price–yield relationship.

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Connection to Advanced Concepts

Duration is the entry point into a family of increasingly sophisticated interest rate risk measures. As you advance in fixed-income analysis, you will encounter extensions that relax the simplifying assumptions of basic (Macaulay/modified) duration. The table below positions the concepts introduced in this lesson relative to their more advanced counterparts.

Introductory Concept	Advanced Extension	What It Adds
Macaulay Duration	Effective Duration	Uses numerical estimation (bump-and-reprice) to capture the impact of embedded options on price sensitivity, rather than assuming fixed cash flows.
Modified Duration (linear)	Duration + Convexity	Adds the second-derivative (curvature) term to the Taylor expansion, significantly improving accuracy for large yield changes.
Parallel shift assumption	Key Rate Duration	Measures sensitivity to changes at specific points on the yield curve (e.g., 2-year, 5-year, 10-year), allowing non-parallel shift analysis.
Single-bond duration	Portfolio Duration & DV01	Aggregates individual duration measures into portfolio-level risk metrics, facilitating hedging and asset–liability management.

From introductory duration to advanced interest rate risk measures

The transition from modified to effective duration is particularly important for fixed-income professionals. Many bonds in practice — mortgage-backed securities, callable corporate bonds, convertible notes — have cash flows that change when interest rates change. Modified duration, which assumes cash flows are fixed, can be misleading for such instruments. Effective duration overcomes this by perturbing the yield curve in both directions and observing the resulting price changes empirically, making it applicable to any security regardless of embedded optionality.

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Looking Ahead

In subsequent lessons, you will derive the convexity adjustment formula, apply key rate duration decomposition, and practice constructing duration-neutral portfolios for hedging applications. The intuition developed here — that duration measures the slope of the price–yield curve — will remain the conceptual anchor for all of these extensions.

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Practice Problems

Question 1: CONCEPTUAL

A 10-year zero-coupon bond and a 10-year bond paying a 7% annual coupon both have the same yield to maturity. Without performing any calculations, which bond has the higher Macaulay duration? Explain the economic reasoning behind your answer.

Show answer

The zero-coupon bond has the higher Macaulay duration. Its duration equals exactly 10 years because all of its cash flow (the face value) arrives at maturity. The 7% coupon bond, by contrast, delivers a substantial portion of its present value through earlier coupon payments, pulling the weighted-average time forward to something less than 10 years. The coupon effect ensures that any bond making interim payments will have a duration shorter than its maturity, and the larger the coupon relative to the face value, the more pronounced this shortening effect becomes.

Question 2: BASIC CALCULATION

A bond has a Macaulay duration of 6.2 years and a yield to maturity of 4% (semiannual compounding). Compute the modified duration.

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Modified duration = Macaulay duration / (1 + y/k). With semiannual compounding, k = 2, so the periodic yield is 4%/2 = 2%, or 0.02. Therefore, D_Mod = 6.2 / (1 + 0.02) = 6.2 / 1.02 = 6.078. The modified duration is approximately 6.08. This means the bond's price is expected to decline by roughly 6.08% for each 1 percentage point (100 basis points) increase in yield to maturity. Notice that modified duration is always slightly less than Macaulay duration because we divide by a factor greater than one.

Question 3: INTERMEDIATE

A 2-year bond with a face value of $1,000 pays a 5% annual coupon and has a yield to maturity of 6%. Compute its Macaulay duration and modified duration. Then estimate the percentage price change if yields decline by 75 basis points.

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First, compute the bond price: PV₁ = 50/1.06 = $47.17, PV₂ = 1050/1.06² = $934.50. Price P = $47.17 + $934.50 = $981.67. Next, compute weighted PVs: 1 × $47.17 = $47.17, 2 × $934.50 = $1,869.00. Sum = $1,916.17. Macaulay duration = $1,916.17 / $981.67 = 1.952 years. Modified duration = 1.952 / (1.06) = 1.841. For a yield decline of 75 bps (Δy = −0.0075): ΔP/P ≈ −1.841 × (−0.0075) = +0.01381, or approximately +1.38%. The estimated price increase is about $13.55, bringing the approximate new price to $995.22. Duration predicts a price rise because yields fell — consistent with the inverse relationship.

Question 4: APPLIED

A pension fund has liabilities with a present value of $500 million and a Macaulay duration of 12 years. The fund manager wants to immunize these liabilities by investing in a portfolio of bonds. She has access to a 5-year zero-coupon bond (duration = 5 years) and a 20-year zero-coupon bond (duration = 20 years). What proportion of the $500 million should be allocated to each bond to achieve immunization?

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Immunization requires matching the portfolio duration to the liability duration of 12 years. Let w represent the weight in the 5-year bond and (1 − w) the weight in the 20-year bond. The portfolio duration equation is: w × 5 + (1 − w) × 20 = 12. Expanding: 5w + 20 − 20w = 12, so −15w = −8, and w = 8/15 ≈ 0.5333. Therefore, approximately 53.33% ($266.67 million) should be invested in the 5-year zero-coupon bond, and 46.67% ($233.33 million) in the 20-year zero-coupon bond. This allocation sets the portfolio's weighted-average duration to 12 years, matching the liability duration and providing first-order protection against parallel yield curve shifts.

Question 5: CRITICAL THINKING

Two bonds have identical modified durations of 8.0. Bond A is a plain-vanilla 15-year corporate bond, while Bond B is a 30-year callable corporate bond. The fund manager remarks that the bonds have 'the same interest rate risk.' Critically evaluate this claim, identifying at least two reasons why identical modified durations may not imply identical risk profiles, and suggest a more appropriate risk measure for Bond B.

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The claim is oversimplified for several important reasons. First, modified duration assumes fixed cash flows, but Bond B is callable — if interest rates fall significantly, the issuer may redeem the bond early, truncating the investor's cash flow stream. The stated modified duration of 8.0 for Bond B was computed assuming current yield levels, but as rates change, the expected cash flows change too, meaning Bond B's effective sensitivity is path-dependent. Second, even if duration is matched at a point in time, the two bonds have very different convexity profiles. Bond A, with no embedded options, exhibits positive convexity (it gains more when rates fall than it loses when rates rise). Bond B, being callable, exhibits negative convexity in the region near the call price — its price appreciation is capped when rates decline because the call option becomes more valuable to the issuer. A more appropriate risk measure for Bond B is effective duration, computed by perturbing the yield curve up and down and observing the model-derived price changes, incorporating the option pricing dynamics. Additionally, option-adjusted spread (OAS) analysis and effective convexity should supplement the assessment.

SUMMARY

Lesson Summary

Summary

Duration is the foundational metric for measuring a bond's sensitivity to interest rate changes. Macaulay duration measures the weighted-average time (in years) until a bond's cash flows are received, using present-value weights derived from the bond's yield to maturity. Modified duration converts this into a direct measure of price sensitivity by dividing by (1 + y/k), yielding the approximate percentage price change per unit change in yield. The key relationship — ΔP/P ≈ −D_Mod × Δy — captures the inverse relationship between bond prices and yields.

Duration is determined by three primary factors: it increases with maturity, decreases with higher coupons, and decreases with higher yields. As a linear (first-order) approximation, it is highly accurate for small yield changes but requires a convexity adjustment for larger shifts. Advanced extensions — effective duration, key rate duration, and portfolio duration — build upon this foundation to handle embedded options, non-parallel yield curve movements, and multi-asset risk management.