Duration of Coupon Bonds
Duration of Coupon Bonds
Coupon bonds always have duration shorter than maturity, because the bond pays you cash along the way—coupons arriving before the maturity date reduce the weighted-average time you wait for full repayment.
Key takeaways
- Duration always less than maturity for coupon bonds (unless maturity is very short)
- Higher coupon rates reduce duration relative to maturity
- Lower coupon rates push duration closer to maturity
- Zero-coupon bonds represent the upper limit (duration = maturity)
- The gap between duration and maturity shrinks for very short-maturity bonds
Why coupons shorten duration
Recall that duration is a weighted average of the timing of cash flows, where each time is weighted by the present value of that cash flow.
For a coupon bond, you receive cash multiple times:
- Year 1: Coupon $C
- Year 2: Coupon $C
- ...
- Year N: Coupon $C + Principal $P
All the $C coupons arrive before the final payment in year N. They pull the weighted average time earlier than year N.
Example: A 10-year bond paying $50 annually.
If all $1,000 of principal arrived in year 10, the weighted average would be at year 10 (duration = 10). But the $50 coupons (total $500 over 10 years) arrive gradually. They pull the weighted average to perhaps 7.5 or 8 years, depending on the yield level.
The fundamental math is: more cash arriving early → shorter weighted average → lower duration.
Numerical example: comparing zero-coupon and coupon
10-year zero-coupon bond:
- Only cash flow: $1,000 at year 10
- Duration: 10.0 years (no coupons to pull forward)
- Maturity: 10 years
- Duration = Maturity
10-year bond, 3% coupon, yielding 3% (trading at par):
- Annual coupons: $30
- Final payment: $1,030 (coupon + principal)
- Macaulay duration: [calculated as weighted average] ≈ 9.0 years
- Maturity: 10 years
- Duration < Maturity
10-year bond, 6% coupon, yielding 3% (trading above par):
- Annual coupons: $60
- Final payment: $1,060
- Macaulay duration: ≈ 8.0 years
- Maturity: 10 years
- Duration < Maturity (and even shorter than the 3% coupon bond)
The higher the coupon, the more cash arrives early, the shorter the duration relative to maturity.
Coupon rates and duration relationship
Let's fix the maturity and yield, and vary only the coupon:
All bonds: 10-year maturity, 4% yield to maturity
| Coupon | Duration | Gap to Maturity |
|---|---|---|
| 0% (zero-coupon) | 9.61 years | 0.39 years |
| 2% | 9.15 years | 0.85 years |
| 4% | 8.48 years | 1.52 years |
| 6% | 7.77 years | 2.23 years |
| 8% | 7.07 years | 2.93 years |
As coupon increases, duration decreases. The zero-coupon bond has the longest duration. The 8% coupon bond has the shortest.
The zero-coupon bond's duration (9.61 years) is less than its maturity (10 years) only because of the yield adjustment (the denominator in modified duration). Macaulay duration for a zero-coupon bond equals maturity exactly.
The extreme cases
Very low coupon (approaching zero):
- Cash flows are nearly all at maturity
- Duration approaches maturity
- Example: A 10-year Treasury paying 0.5% coupon has duration ≈ 9.8 years
Very high coupon:
- Cash flows are heavily weighted toward early years
- Duration is much shorter than maturity
- Example: A 10-year bond paying 10% coupon has duration ≈ 6.5 years
Mid-range coupons:
- Duration typically 60–80% of maturity for typical bonds
Practical implications for bond selection
Coupon bonds' short duration (relative to maturity) is why coupon bonds are less volatile than zero-coupon bonds of the same maturity.
If you buy a 10-year 5% coupon bond versus a 10-year zero-coupon bond, expecting to hold both one year:
- The zero-coupon bond has duration 10 years; a 1% yield rise costs ≈10%
- The coupon bond has duration ≈7.5 years; a 1% yield rise costs ≈7.5%
The coupon bond is less risky if rates rise. But it also provides less upside if rates fall.
This is why investors' maturity choices don't directly match duration choices. A 5-year investor might buy a 7-year bond with 5-year duration, getting slightly higher yield while staying roughly matched to their time horizon.
Coupon bonds and reinvestment risk
There's a subtle trade-off. Coupon bonds' shorter duration means less interest rate risk—if rates rise, your coupons are less affected. But you face reinvestment risk: you must reinvest coupons at uncertain rates.
If you receive a $50 coupon in year 1, you must reinvest it for the remaining 9 years. If rates have fallen, you reinvest at lower rates. If rates have risen, you reinvest at higher rates.
A zero-coupon bond has no reinvestment risk (no interim coupons to reinvest) but maximum interest rate risk (all cash at one date, vulnerable to yield moves).
This is why liability-driven investment strategies often use zero-coupon bonds (STRIPs): they eliminate reinvestment risk for exact-date liabilities.
Duration of bonds with varying coupons
In a portfolio, different bonds have different coupons and thus different durations. A bond portfolio duration is the weighted average of individual bond durations.
Example:
- $3 million of 10-year bonds with 2% coupon: duration 9.3 years
- $7 million of 10-year bonds with 6% coupon: duration 7.5 years
Portfolio duration: = ($3M × 9.3 + $7M × 7.5) / ($3M + $7M) = ($27.9M + $52.5M) / $10M = $80.4M / $10M = 8.04 years
The portfolio's duration is less than a simple average of the two (8.4 years) because the larger position (7% coupon, shorter duration) dominates.
Coupon bonds in index funds
Bond index funds (BND, AGG, BNDX) hold thousands of bonds with varying coupons. The portfolio's average coupon and average duration reflect the mix.
AGG (U.S. Aggregate Bond Index) has roughly:
- Average maturity: 7.5 years
- Average duration: 5.5 years
- Average coupon: 4.5%
The duration is well below maturity because of the mixed composition—many coupon-paying bonds pull the weighted average forward.
Compare this to a pure long-duration bond index or fund that might have:
- Average maturity: 15–20 years
- Average duration: 10–15 years
- More skew toward longer, lower-coupon bonds
The mathematics of duration vs. maturity
The precise relationship between duration and maturity for coupon bonds is complex, involving Macaulay duration, modified duration, effective duration, and the yield level.
But the intuitive rule is simple: coupon bonds' duration < maturity, always. How much less depends on coupon size and yield.
Coupon timing
Bond duration simplification
For quick mental estimates, practitioners often use rules of thumb:
For plain-vanilla bonds:
- Duration ≈ 75% of maturity (rough average)
- For 10-year bonds: duration ≈ 7–8 years
- For 5-year bonds: duration ≈ 4–4.5 years
For low-coupon bonds:
- Duration ≈ 90% of maturity or higher
For high-coupon bonds:
- Duration ≈ 60–70% of maturity
These are rules of thumb; precise calculations require actual coupon and yield data.
Next
Coupon bonds have shorter duration than maturity because cash arrives throughout the bond's life. When you combine multiple bonds into a portfolio, their durations combine—the topic of the next article, which explores how portfolio duration is calculated and managed.