Duration and Yield Relationship
Duration and Yield Relationship
For a given bond, duration decreases as yield to maturity increases and increases as yield decreases. A bond trading at high yield has shorter duration than the same bond trading at low yield.
Key takeaways
- Higher yield → lower duration (inverse relationship)
- Lower yield → higher duration
- The effect is usually smaller than the coupon effect but is always present
- At very high yields, duration can become quite short
- This relationship is built into the modified duration formula: Duration_modified = Duration_Macaulay / (1 + y)
The intuition: present-value weighting shifts
Duration is a weighted average of cash flow timings, where weights are the present values of each cash flow relative to total bond price.
When yields rise:
- Present values of all future cash flows decline
- But near-term cash flows decline proportionally less than distant cash flows (because the discounting is less severe)
- Near-term cash flows become a relatively larger fraction of total value
- The weighted average time shifts earlier
- Duration decreases
When yields fall:
- Present values of all future cash flows rise
- Distant cash flows rise proportionally more than near-term cash flows
- Distant cash flows become a relatively larger fraction of total value
- The weighted average time shifts later
- Duration increases
Example: A 10-year bond paying $50 annually plus $1,000 at year 10.
At 3% yield:
- Year 1 coupon: PV = $50 / 1.03 = $48.54
- Year 10 final payment: PV = $1,050 / 1.03^10 = $779.55
- Total PV (bond price): ≈ $873.52 (simplified; missing intermediate coupons)
- Weights: Year 1 is 5.6% of value; Year 10 is 89.3% of value
- Weighted average: heavily weighted toward year 10
- Duration: ≈ 8.2 years (longer)
At 6% yield:
- Year 1 coupon: PV = $50 / 1.06 = $47.17
- Year 10 final payment: PV = $1,050 / 1.06^10 = $585.62
- Total PV (bond price): ≈ $735.72
- Weights: Year 1 is 6.4% of value; Year 10 is 79.6% of value
- Weighted average: still weighted toward year 10, but less extremely
- Duration: ≈ 7.6 years (shorter)
As yield rises from 3% to 6%, duration falls from 8.2 to 7.6 years. The shift is smaller than the coupon effect, but consistent.
Mathematical foundation: the modified duration formula
Modified duration encodes the yield relationship:
Modified duration = Macaulay duration / (1 + y)
As y (yield as a decimal) increases, the denominator grows, and modified duration shrinks.
Example: Macaulay duration 8.0 years
- At 2% yield: Modified = 8.0 / 1.02 = 7.84 years
- At 4% yield: Modified = 8.0 / 1.04 = 7.69 years
- At 6% yield: Modified = 8.0 / 1.06 = 7.55 years
- At 8% yield: Modified = 8.0 / 1.08 = 7.41 years
The (1 + y) denominator directly reduces modified duration as yield increases.
Numerical comparison across yield levels
10-year bond, 4% coupon
| Yield | Macaulay Duration | Modified Duration | Bond Price |
|---|---|---|---|
| 2% | 9.22 | 9.04 | $1,196.36 |
| 3% | 9.17 | 8.91 | $1,112.79 |
| 4% | 9.12 | 8.77 | $1,000.00 |
| 5% | 9.07 | 8.64 | $922.78 |
| 6% | 9.03 | 8.52 | $858.96 |
As yield increases from 2% to 6%:
- Macaulay duration falls only slightly (9.22 to 9.03), about 0.19 years
- Modified duration falls more noticeably (9.04 to 8.52), about 0.52 years
- Bond price falls significantly (from $1,196 to $859)
The Macaulay duration barely moves because the coupon-to-maturity ratio (4% coupon on a 10-year bond) dominates the calculation. The yield effect is smaller than the coupon effect.
But the modified duration, which includes the yield adjustment, does fall meaningfully.
High-yield bonds: extreme case
In high-yield (junk) bond markets, yields can be 8%, 10%, or higher. These bonds' durations are compressed.
Example: 10-year high-yield bond, 9% coupon, yielding 8%
- Large coupons: $90 annually
- High yield: 8%
- Expected duration: ≈ 6.2 years (much shorter than maturity)
Compare to a similar 10-year bond with 3% coupon, yielding 3%:
- Small coupons: $30 annually
- Low yield: 3%
- Expected duration: ≈ 9.2 years
Both mature in 10 years. The high-yield bond (9% coupon, 8% yield) has duration roughly 6.2 years. The low-coupon, low-yield bond has duration 9.2 years. Duration differs by 3 years—driven by both coupon (high vs. low) and yield (high vs. low).
Practical example: market environment shifts
In early 2021, when yields were very low (1–2%):
- Bonds had relatively long duration
- BND (Vanguard Total Bond Market) had duration ≈ 6.5 years
By late 2023, after aggressive Fed hikes, yields were high (4–5%):
- The same bonds (BND) had duration ≈ 5.2 years
- The higher yield environment compressed durations
A fixed portfolio duration decreased not because holdings changed, but because yields rose.
Implication: duration is environment-dependent
Duration is not a fixed property of a bond. It depends on the current yield.
When you say "a 10-year bond has 8-year duration," you're implicitly assuming a particular yield. If yields have changed since your analysis, duration has also changed.
For bonds held in a portfolio:
- In low-rate environments (2020–2021), portfolios tend to have longer duration
- In high-rate environments (2022–2025), the same physical bonds have shorter duration
This is why portfolio managers rebalance. If your target duration is 5 years and yields fall, your portfolio's actual duration might rise to 5.5 years (holding the bonds). You'd need to sell longer-duration bonds and buy shorter-duration ones to maintain your target.
The yield effect vs. coupon effect
Which is stronger—coupon or yield?
For a 10-year bond:
- Coupon effect: varying coupon from 2% to 8% (all at same yield) changes duration by ≈0.9 years
- Yield effect: varying yield from 2% to 8% (all at same coupon) changes duration by ≈0.3–0.5 years
The coupon effect is typically 2–3× larger than the yield effect.
However, both effects combine. A low-coupon, low-yield bond has longest duration. A high-coupon, high-yield bond has shortest duration. The two effects reinforce each other.
Callable bonds and the yield effect
For callable bonds, the yield effect is complicated by the embedded call option.
When yields rise, the call option becomes less likely to be exercised (the issuer won't call if the bond is worth less than the call price). The bond behaves like a regular bond, and duration is stable or increases slightly.
When yields fall, the call option becomes more likely to be exercised. The bond's duration shortens dramatically because the price is capped at the call price. The normal inverse yield-duration relationship is disrupted.
This is why effective duration is necessary for callable bonds—the numerical approximation captures the option effects correctly.
Real-world illustration: different rate environments
2016 environment (low yields, near-zero Fed funds rate):
- 10-year Treasury yield: 2.4%
- 10-year Treasury modified duration: ≈ 9.0 years
- High interest rate sensitivity
2019 environment (moderate yields, Fed holding steady):
- 10-year Treasury yield: 2.0%
- 10-year Treasury modified duration: ≈ 9.1 years
- Similar to 2016
2022 environment (high yields, aggressive Fed hikes):
- 10-year Treasury yield: 4.0%
- 10-year Treasury modified duration: ≈ 7.5 years
- Much lower interest rate sensitivity (shorter duration)
The same Treasury bond (if we could compare it over time, though Treasuries are issued with different coupons) would have different duration in different rate environments.
The economic interpretation
From a present-value perspective, the yield relationship makes intuitive sense.
In a high-yield environment (say, 6%), future cash flows are heavily discounted. The present value of a dollar received in year 5 is much less than the present value of a dollar received in year 1. Near-term cash flows dominate the valuation and weighting.
In a low-yield environment (say, 2%), future cash flows are lightly discounted. Distant payments are nearly as valuable (in present-value terms) as near-term payments. Distant cash flows get higher weight in the duration calculation.
This is why bonds' duration lengths follow the yield curve. In normal upward-sloping yield curves, longer-maturity bonds have higher yields, which compresses their duration slightly. In inverted curves (short-term yields higher than long-term yields), duration patterns are inverted.
Yield relationship flowchart
Next
We've now covered the three key duration relationships: with coupon, maturity, and yield. Together, they explain duration behavior across all bond types and market environments. In the next (final) article of the section, we move beyond duration to convexity—the curvature in the price-yield relationship that becomes important for large rate moves. Duration is linear; convexity captures the nonlinearity.