Duration as Price Sensitivity
Duration as Price Sensitivity
From a trader's and risk manager's perspective, duration is a single number that estimates how much a bond's price rises or falls (in percentage terms) for every 1% change in interest rates.
Key takeaways
- A bond with 5-year duration loses roughly 5% when yields rise 1%; gains roughly 5% when yields fall 1%
- This relationship comes directly from the mathematics of present-value calculations and bond pricing
- Longer-duration bonds are more sensitive to rate changes; shorter-duration bonds are less sensitive
- Modified duration and effective duration are the metrics traders and fund managers use operationally
- Understanding duration as price sensitivity is essential for portfolio risk management and hedging
The bond pricing formula and its sensitivity
When you buy a bond, you pay the present value of all future cash flows, discounted at the yield to maturity:
Bond price = CF₁/(1+y) + CF₂/(1+y)² + ... + CFₙ/(1+y)ⁿ
Where CF is cash flow and y is the yield to maturity.
Now, what happens if yields rise from 4% to 5% (a 1% increase)?
All the denominators get larger: (1.04) becomes (1.05), (1.04)² becomes (1.05)², and so on. Larger denominators mean smaller present values. The bond's price falls.
How much does it fall? That depends on the magnitude and timing of the cash flows. A bond with cash flows concentrated far in the future (long duration) experiences much larger present-value declines when yields rise, because the discounting effect compounds over many years.
A bond with most cash flows arriving soon (short duration) experiences smaller declines, because the discounting effect operates over fewer years.
This is where duration quantifies the sensitivity. The mathematical relationship turns out to be linear (approximately, for small yield changes): duration in years equals percentage price change per 1% yield move.
The mechanics of price decline
Let's walk through a concrete example. Consider a 5-year bond paying 4% coupon, currently trading at par ($1,000) with 4% yield.
If yields stay at 4%:
- Year 1: $40 coupon, discounted at 4%: $40/1.04 = $38.46
- Year 2: $40, discounted: $40/1.04² = $36.97
- Year 3: $40, discounted: $40/1.04³ = $35.56
- Year 4: $40, discounted: $40/1.04⁴ = $34.19
- Year 5: $1,040, discounted: $1,040/1.04⁵ = $854.80
- Total: $1,000 (par, as expected)
If yields rise to 5%:
- Year 1: $40/1.05 = $38.10
- Year 2: $40/1.05² = $36.28
- Year 3: $40/1.05³ = $34.60
- Year 4: $40/1.05⁴ = $32.95
- Year 5: $1,040/1.05⁵ = $815.37
- Total: $957.30 (fallen from $1,000)
Price decline: ($1,000 - $957.30) / $1,000 = 4.27%
For a 1% yield rise, the bond lost 4.27%. This bond's duration is approximately 4.27 years. (More precisely, it's slightly less because of convexity, but duration captures the first-order effect.)
Duration and portfolio mark-to-market
For fund managers and traders, duration is a risk metric. It tells you how much your position's value will fluctuate if rates move.
If you manage a $100 million bond portfolio with average duration 4.5 years:
- A 1% yield rise costs you: $100 million × 4.5% = $4.5 million
- A 1% yield fall gains you: $100 million × 4.5% = $4.5 million
- A 0.5% yield rise costs: $2.25 million
- A 0.25% yield rise costs: $1.125 million
This is the daily tool. Portfolio managers track:
- The duration of their bond holdings
- The duration of their liabilities or benchmarks
- The duration gap (mismatch) between assets and liabilities
- How duration will shift if rates move
Constructing portfolios by duration target
Asset managers construct portfolios with specific duration targets. The BND index (Vanguard Total Bond Market), for example, has roughly 5.5 years duration, reflecting the weighted-average duration of all eligible U.S. bond holdings.
If you believe rates will rise, you might underweight BND (which has 5.5-year duration) and instead buy a shorter-duration bond fund like a 1–3 year Treasury fund (with ~1.5 year duration). You're reducing your interest rate risk.
If you believe rates will fall, you might overweight longer-duration bonds, amplifying your price gains.
AGG (the Aggregate Bond Index), IEF (intermediate Treasury ETF), TLT (long-term Treasury), HYG (high-yield corporate): each has a different average duration, offering different interest rate sensitivity.
Duration and leverage
Duration also drives leverage decisions. A bond trader might use leverage to amplify returns. If 1-year Treasury bills yield 5% and you can borrow at 4.5%, you can earn 0.5% spread by borrowing and buying Treasury bills.
But what if the Fed cuts rates? Your leverage amplifies losses. Duration quantifies this leverage risk.
A leveraged position in bonds with 5-year duration becomes very sensitive to rate moves. If rates fall 2%, the leverage amplifies your gains. If rates rise 2%, the leverage amplifies your losses. Leverage × duration = total interest rate risk.
Hedging with duration
Large bond investors use duration to hedge interest rate risk.
Example: A pension fund has liabilities (promised pension payments) that are effectively like bonds. The liabilities might have 10-year duration. To hedge interest rate risk, the pension fund wants bond assets with 10-year duration, creating a duration match.
If bond yields fall, both the pension fund's liabilities (present value of future payments) and its assets rise in value. The hedge is effective.
Duration is the metric that makes this possible. By matching durations, you immunize the portfolio against most interest rate moves.
Why duration is approximately linear
The linearity of the duration–price relationship (duration in years = percentage price change per 1% yield move) comes from calculus and the structure of the present-value formula.
Mathematically, the "duration" is the elasticity of the bond price with respect to yields. It's the first derivative of the price function with respect to yield, scaled appropriately. For small yield changes, this linear approximation is excellent.
For large yield changes (2% or more), convexity—the curvature in the price–yield relationship—becomes significant. But for day-to-day risk management and most market conditions, duration is sufficiently accurate.
Duration across market conditions
Duration's power as a sensitivity metric is that it's consistent across market states.
In 2019, when the Fed was cutting rates, a 5-year duration bond gained roughly 5% for every 1% rate fall, providing clear hedging and upside exposure.
In 2022, when the Fed was hiking aggressively, a 5-year duration bond lost roughly 5% for every 1% rate rise, which is why bond funds suffered significant losses.
In 2024–2025, duration remains the risk metric for bond strategies, regardless of the rate environment.
This consistency makes duration invaluable for multi-year, multi-regime analysis.
Price sensitivity flowchart
Practical risk monitoring
Bond portfolio managers track duration constantly. Bloomberg terminals, fund fact sheets, and portfolio management systems all display:
- Current portfolio duration
- Duration benchmark (for active managers)
- Duration sensitivity: "Dollar value of 1 basis point" (DV01), which is just duration expressed in dollar terms
If your $100 million portfolio has 4-year duration, the DV01 is: DV01 = $100 million × 4% × 0.0001 = $4,000 per basis point (0.01%)
This means a 10-basis-point move in yields costs or gains $40,000.
Duration across fixed income
Duration applies universally:
- Treasuries (TLT, IEF, SHV): Duration varies from under 1 year (short-term) to 20+ years (long-term)
- Corporates (LQD, HYG): Duration typically 5–7 years for investment-grade, 3–5 years for high-yield (due to call options shortening it)
- Municipals (MUB): Similar to Treasuries; duration varies by maturity mix
- Mortgage-backed (MBB): Typically 4–6 years, with embedded refinance options
- International bonds (BNDX): Duration varies by country; 5–7 years for developed markets
All these are risk-managed using the duration metric.
Next
Duration as price sensitivity is the trader's daily tool. But the relationship between price and yield is not perfectly linear; there's curvature. Convexity—the shape of the price–yield curve—explains the gap between linear duration estimates and actual price changes, especially for large rate moves. That's the foundation of the next chapter's exploration.