Government Bond Duration and Interest Rate Risk

Bond duration measures how sensitive a government bond’s price is to changes in interest rates. Modified duration—the most useful version for investors—tells you the percentage price change for each 1% move in yield. A 2-year Treasury might have a duration of 1.9 years; a 10-year note, around 8.5 years. Rate rises hurt long-duration bonds much more than short ones.

What duration measures: the time-value summary

Macaulay duration is the weighted average time (in years) until you receive all your cash flows from a bond—coupons and principal repayment. A bond that pays $100 annually for two years and then $1,100 at maturity has most of its cash near the end, so its duration is close to, but slightly less than, two years. A 10-year bond stretches that average much further into the future.

Modified duration takes that concept and converts it into a price-sensitivity metric. It answers: “If yields rise 1%, what percentage does my bond’s price fall?”

The relationship is direct and negative: higher duration means bigger price swings when rates move.

How to think about it visually

Imagine two government bonds:

1. A 2-year Treasury yielding 4.0% with a modified duration of 1.9.
2. A 10-year Treasury yielding 4.5% with a modified duration of 8.5.

Suppose the Federal Reserve signals tighter policy and yields jump 1.0% across the board (all rates rise from their current levels by exactly 1 percentage point).

• 2-year bond: Price falls ≈ 1.9% (a relatively modest loss).
• 10-year bond: Price falls ≈ 8.5% (a much steeper loss).

If you own $100,000 face value of the 2-year, you lose roughly $1,900. If you own $100,000 face value of the 10-year, you lose roughly $8,500. The long-duration bond is far more vulnerable to rate moves.

Why longer bonds have higher duration

Two factors explain this. First, a longer-maturity bond pays most of its cash flow far in the future. When rates rise, that distant cash becomes worth less in today’s dollars—a bigger hit than a short-dated bond.

Second, you’re locked into an old, lower coupon for longer. If you own a 10-year Treasury yielding 4%, and new 10-year Treasuries suddenly yield 5%, you can’t “reinvest” your old coupon into a higher rate for another 9.5 years (you’re stuck with 4%). With a 2-year bond in the same scenario, you get back your principal in two years and can reinvest at 5% for the remaining eight years. That earlier exit reduces rate sensitivity.

A concrete example: the 2-year versus 10-year

Let’s compare two real scenarios using realistic numbers.

Scenario A: A 2-year Treasury

• Par: $10,000
• Coupon: 4.0% annual (paid semi-annually, $200 each)
• Maturity: 2 years
• Current yield to maturity: 4.0%
• Current price: $10,000 (trading at par)
• Modified duration: 1.94 years

Scenario B: A 10-year Treasury

• Par: $10,000
• Coupon: 4.5% annual (paid semi-annually, $225 each)
• Maturity: 10 years
• Current yield to maturity: 4.5%
• Current price: $10,000 (trading at par)
• Modified duration: 8.53 years

Now, rates rise 1.0% instantly across all maturities.

2-year bond after rate rise:

• New yield: 5.0%
• New price ≈ $10,000 × (1 − 0.0194) = $9,806
• Loss: $194, or 1.94%

10-year bond after rate rise:

• New yield: 5.5%
• New price ≈ $10,000 × (1 − 0.0853) = $9,147
• Loss: $853, or 8.53%

The 10-year loss is more than four times larger in percentage terms, despite the same 1% rate move. This is the power—and peril—of duration.

The inverse relationship: yields fall, bonds rally harder with longer duration

Duration is a two-way street. If yields fall 1%, the same calculation applies with a plus sign:

If the 2-year yield drops 1%:

• New price ≈ $10,000 × (1 + 0.0194) = $10,194
• Gain: $194, or 1.94%

If the 10-year yield drops 1%:

• New price ≈ $10,000 × (1 + 0.0853) = $10,853
• Gain: $853, or 8.53%

Long-duration bonds magnify gains when rates fall. This is why investors “reach for duration” in periods of falling rates or recession fears. But it cuts both ways: in a rising-rate environment, duration is your enemy.

Duration varies across the curve

The modified duration of government bonds depends on three things:

1. Maturity: Longer maturities have higher durations, but not linearly. A 30-year Treasury doesn’t have 15 times the duration of a 2-year; it might have 10–12 times. This is because the yield denominator in the modified duration formula is larger for longer bonds.

2. Coupon rate: Higher coupons mean lower duration. A 30-year Treasury yielding 5% has lower duration than a 30-year Treasury yielding 3% because you’re collecting cash faster. Zero-coupon bonds (like very short-maturity T-bills) have durations close to their maturity (since all cash comes at the end).

3. Yield level: Higher yields reduce duration. A bond yielding 6% has lower duration than the same bond yielding 3%. This is purely a mathematical effect: the denominator in the modified duration formula grows with yield.

Using duration to manage interest rate risk

Portfolio managers use duration to control rate sensitivity. If a manager expects rates to fall, she might increase her portfolio’s average duration (buy longer bonds) to amplify the expected gains. If she expects rates to rise, she might shorten duration (shift to shorter bonds, bond funds, or floating-rate notes) to limit losses.

Duration also helps answer “what-if” questions: “If the Fed raises rates 0.5%, what’s my bond portfolio’s expected loss?” Multiply your portfolio’s weighted average duration by 0.5%.

The limits of duration: convexity

Duration is a linear approximation. It works well for small rate moves (±0.5%) but breaks down for large moves. A bond’s price-yield relationship is actually curved (convex), not straight, so a 2% rate drop produces a larger gain than a 2% rate rise produces a loss, even with the same duration.

This is convexity, and it’s particularly important for long-duration bonds. But for most daily use, duration alone is enough to gauge rate risk.

See also