Duration and Maturity Relationship
Duration and Maturity Relationship
Duration increases with maturity, but the relationship is nonlinear: very long-maturity bonds have only slightly longer duration than intermediate-maturity bonds, because distant cash flows are heavily discounted.
Key takeaways
- Duration always increases when maturity increases (all else equal)
- The relationship is nonlinear—adding 10 years to maturity doesn't add 10 years to duration
- Zero-coupon bonds show this most clearly: a 20-year zero has 20-year duration, a 30-year zero has 30-year duration, but the gap matters less in present-value terms
- For coupon bonds, a 30-year bond might have only 8–10 year duration, not 30 years
- Duration asymptotes (approaches a limit) as maturity extends indefinitely
The mathematics: why duration grows slower than maturity
Duration is a weighted average of cash flow timings, weighted by present value. As maturity extends, future cash flows are discounted exponentially.
Present value of a cash flow in year N = CF / (1 + y)^N
As N gets large (20, 30, 50 years), the denominator grows exponentially. The present value shrinks dramatically.
Example: $100 cash flow received in different years, at 4% yield:
- Year 5: PV = $100 / 1.04^5 = $82.19
- Year 10: PV = $100 / 1.04^10 = $67.56
- Year 20: PV = $100 / 1.04^20 = $45.64
- Year 30: PV = $100 / 1.04^30 = $30.83
- Year 50: PV = $100 / 1.04^50 = $7.11
A cash flow in year 50 is worth only 7% as much (in present-value terms) as the same cash flow in year 5. It barely affects the weighted average time calculation.
This is why a 50-year bond's duration is typically not that much longer than a 30-year bond—the distant cash flows hardly matter.
Numerical examples: duration vs. maturity
Zero-coupon bonds:
| Maturity | Duration | Gap |
|---|---|---|
| 5 years | 5.00 | 0 |
| 10 years | 10.00 | 0 |
| 20 years | 20.00 | 0 |
| 30 years | 30.00 | 0 |
For zero-coupon bonds, duration equals maturity exactly (Macaulay duration; modified duration is slightly less).
Coupon bonds (4% coupon, 4% yield):
| Maturity | Duration | Gap | Duration / Maturity |
|---|---|---|---|
| 5 years | 4.44 | 0.56 | 89% |
| 10 years | 8.12 | 1.88 | 81% |
| 20 years | 13.63 | 6.37 | 68% |
| 30 years | 16.88 | 13.12 | 56% |
Notice: Maturity doubles from 5 to 10 years, but duration only increases from 4.44 to 8.12 (less than doubling). Maturity triples from 10 to 30 years, but duration only increases from 8.12 to 16.88 (roughly doubling).
The duration / maturity ratio shrinks as maturity extends. A 5-year bond's duration is 89% of maturity; a 30-year bond's duration is only 56% of maturity.
Why duration and maturity diverge
For coupon bonds, duration is always less than maturity because you receive interim cash flows. Those coupons pull the weighted average earlier than the maturity date.
The longer the bond, the greater the gap. Here's why:
Coupons arrive every year. By year 20, you've received 20 years of coupons—a substantial portion of total cash. The final principal (arriving in year 30) is discounted heavily and represents a small fraction of total present value.
For a 5-year bond, coupons are 50% of total cash value, and principal is 50%. For a 30-year bond, coupons are roughly 60–70% of total present value, and principal is only 30–40%. The larger proportion of early payments (coupons) pulls the weighted average further below maturity.
The asymptotic behavior
As maturity approaches infinity, duration approaches a limit.
For a perpetuity (a bond that never matures, paying coupons forever), the duration is:
Duration_perpetuity = (1 + y) / y
At 4% yield: Duration = 1.04 / 0.04 = 26 years
This is the limit. No bond—no matter how long its maturity—can have duration longer than this perpetuity duration for that yield level.
A 50-year bond at 4% yield has duration perhaps 17–18 years. A 100-year bond might have duration 19–20 years. A 1000-year bond would be even closer to 26 years (the perpetuity limit), but never exceeding it.
This is a profound mathematical fact: very long bonds are not that much riskier than moderately long bonds, in duration terms.
Practical implications
30-year Treasuries have duration around 17–18 years, not 30 years. While they're far more sensitive to rates than intermediate bonds (5–10 year duration), they're not 3× more sensitive.
Ultra-long bonds (50-year corporates, rare 100-year bonds) offer only modestly more duration exposure than 30-year bonds. The extreme maturity offers limited additional rate sensitivity because future cash flows are so heavily discounted.
This is why investors seeking maximum duration exposure focus on bonds in the 10–20 year maturity range, not ultra-long bonds. The return enhancement from extreme maturity is often not worth the additional liquidity risk and other complications.
The duration-maturity relationship in practice
Bond investors monitor duration, not maturity, because duration captures what actually matters: interest rate sensitivity and average time to repayment.
Two portfolios with identical maturity structure but different coupon structures will have different durations and thus different interest rate risk.
Example:
- Portfolio A: Mix of low-coupon Treasuries, average maturity 10 years, average duration 9 years
- Portfolio B: Mix of high-coupon corporates, average maturity 10 years, average duration 7 years
Both have 10-year maturity. Portfolio A is significantly more rate-sensitive (duration 9 vs. 7 years). An investor seeking short duration would choose Portfolio B despite the identical maturity.
Duration vs. maturity in ladder strategies
Bond ladders—buying bonds of various maturities (1 year, 2 year, 3 year, etc.) to provide cash inflows—are sometimes constructed by maturity and sometimes by duration.
A maturity ladder is simple: buy a bond expiring each year. This is easy to manage but leaves you with uneven interest rate exposure (the 1-year bond has short duration; the 10-year bond has long duration).
A duration ladder (if possible) would aim for more even duration spacing. But because coupons and yields differ, creating a perfect duration ladder is complex.
Most practitioners use maturity ladders for simplicity, accepting that interest rate sensitivity varies.
Relationship visualization
Maturity and duration relationships across bond types
Treasuries (various maturities, low coupons ~2%):
- 5-year: ~4.7 years duration
- 10-year: ~8.2 years duration
- 20-year: ~13.5 years duration
- 30-year: ~16.8 years duration
Each 10-year maturity increase adds less duration. 5→10 adds 3.5 years; 10→20 adds 5.3 years; 20→30 adds 3.3 years.
Corporate bonds (various maturities, higher coupons ~5%):
- 5-year: ~4.4 years duration
- 10-year: ~8.0 years duration
- 20-year: ~12.8 years duration
- 30-year: ~15.5 years duration
Similar pattern: each additional 10 years of maturity adds progressively less duration.
Next
We've explored duration in relation to coupon, maturity, and time. We've also touched on yields. In the final foundational article of this section, we examine how yields themselves affect duration—why a bond with higher yield has shorter duration than a similar bond with lower yield.