Duration of a Zero-Coupon Bond
Duration of a Zero-Coupon Bond
For a zero-coupon bond, duration equals maturity exactly, because there are no interim cash flows—you receive all principal and accrued interest at one date only.
Key takeaways
- Zero-coupon bond duration = maturity (e.g., a 10-year zero has 10-year duration)
- All cash flows arrive at a single point: the maturity date
- Zero-coupon bonds are the most interest-rate-sensitive bonds of their maturity length
- A 10-year zero loses roughly 10% when yields rise 1%; a 10-year coupon bond loses less
- STRIP securities are real-world zero-coupon bonds created by stripping coupon payments from Treasuries
Why zero-coupon duration is simplest
A zero-coupon bond has no coupons. You buy it at a discount, hold it, and receive par value at maturity. The formula for duration simplifies dramatically.
For a regular bond, duration is a weighted average of years, with each year weighted by the proportion of present value. A 10-year bond with 5% coupons returns some cash in year 1, some in year 2, and so on, all the way to year 10. The weighted average is somewhere between year 1 and year 10—usually around 7–8 years, depending on the coupon rate.
For a zero-coupon bond, there's only one cash flow: the entire face value at maturity. There's nothing to weight against. The average time you wait is simply the maturity date.
Example: A 10-year zero-coupon bond has:
- Present value of cash flows: $X in year 10
- No cash in years 1–9
- Weighted average time: 10 years
- Duration = 10 years
Comparing zero-coupon to coupon bonds
This highlights the stark difference between zero-coupon and coupon bonds of the same maturity.
10-year zero-coupon bond:
- Duration: 10.0 years
- You get all $1,000 at year 10
- Price sensitivity: ≈10% per 1% yield change
10-year 5% coupon bond:
- Duration: ≈7.5 years (coupon payments pull it forward)
- You get $50 each year plus $1,000 in year 10
- Price sensitivity: ≈7.5% per 1% yield change
10-year 8% coupon bond:
- Duration: ≈6.5 years (higher coupons pull it forward more)
- You get $80 each year plus $1,000 in year 10
- Price sensitivity: ≈6.5% per 1% yield change
All three mature in 10 years. But the zero-coupon bond's duration is longest because there are no interim cash flows to reduce the wait time. The zero-coupon bond is also the most volatile, swinging 10% for every 1% yield move, while the higher-coupon bond swings only 6.5%.
This is why zero-coupon bonds appeal to long-term investors willing to accept volatility: they offer pure duration exposure. And it's why they're riskier in rising-rate environments—your entire return depends on one date, and that future cash flow is heavily discounted if rates rise.
The extreme case: sensitivity to rates
Imagine you buy a 20-year zero-coupon bond for $1,000, expecting to receive $4,500 at year 20 (reflecting a low current yield).
Now yields rise 1%. The expected $4,500 is discounted at a higher rate:
- New price ≈ $4,500 / (1.05)²⁰ (assuming new yield is 5%)
At the original yield, the price is $1,000:
- Original price = $4,500 / (1.04)²⁰ = $1,000
The new price is roughly $1,700 lower—a 17% decline from a 1% yield increase.
Duration of 17 years is roughly correct. (Precisely, it's (1 + 0.04) × 16 = 16.64 years due to the yield adjustment, but the order of magnitude is clear.)
A 20-year coupon bond would lose much less. Its duration is typically 12–14 years, not 17–18. The interim coupons reduce the duration significantly.
Zero-coupon bonds in practice: STRIP securities
The most common zero-coupon bonds are STRIP (Separate Trading of Registered Interest and Principal) securities, created by financial institutions from U.S. Treasury bonds.
A Treasury bond paying coupons is "stripped" into separate pieces:
- Individual coupon payments (each becomes a zero-coupon security maturing on a coupon date)
- The principal payment (becomes a zero-coupon security maturing at original maturity)
For example, a 10-year Treasury paying $30 semiannually plus $1,000 at maturity can be separated into 21 securities: 20 coupon strips (each $30, maturing every 6 months) and 1 principal strip ($1,000, maturing in 10 years).
Investors buy these pieces separately, creating a market for zero-coupon securities at every maturity from 6 months to 30 years.
A 5-year STRIP has duration 5 years. A 10-year STRIP has duration 10 years. It's the purest form of duration—no coupons to muddy the calculation.
Inflation-protected zero-coupon bonds?
Some issuers create zero-coupon bonds indexed to inflation (not common in the U.S., but existing in other countries). These are still zero-coupon—no interim payments—but the maturity-date payment is adjusted for inflation.
Duration still equals maturity, but the maturity-date amount is uncertain (depends on inflation through the holding period). This adds inflation risk on top of interest rate risk.
Pricing zero-coupon bonds
A zero-coupon bond selling for $600 with 5 years to maturity, assuming $1,000 face value:
Implied yield to maturity: $1,000 = $600 × (1 + y)⁵ (1 + y)⁵ = 1.6667 y ≈ 10.84%
The bond's yield is 10.84%, and its duration is exactly 5 years.
If yields fall to 10%, the new price is: $1,000 / (1.10)⁵ = $620.92
Price gain: ($620.92 - $600) / $600 = 3.49%
For a 0.84% yield decline, the price gained 3.49%. Duration estimate: 3.49% / 0.84% ≈ 4.16 years. Close to the 5-year duration (the gap is due to convexity; for small moves, the error is small).
Zero-coupon bonds in portfolios
Investors use zero-coupon bonds for specific purposes:
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Liability matching: A pension fund needing $5 million in exactly 10 years buys 10-year STRIPs worth $5 million face value. It locks in the timing and amount—no reinvestment risk.
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Speculation on rates: Traders who expect yields to fall buy long-dated STRIPs, maximizing duration exposure. A 20-year STRIP amplifies price gains if rates fall 1% (roughly 20% gain).
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Tax efficiency: Some zero-coupon securities are structured for tax-deferred accounts where accrued interest isn't taxed annually.
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Simplicity of duration: Researchers and educators use zero-coupon bonds because duration calculation is trivial (duration = maturity).
Education example
Zero-coupon bonds are often the first example when teaching duration. They make the concept concrete:
"A 5-year zero-coupon bond has 5-year duration because you wait 5 years for your money. A 5-year coupon bond has shorter duration—maybe 4.2 years—because you get some coupons along the way, shortening your average wait."
This simple, intuitive example sets the stage for understanding more complex bonds.
Pure duration expression
Zero-coupon bond volatility illustration
Consider three bonds, all yielding 4%, comparing price changes when yields rise 1%:
5-year zero-coupon bond:
- Original price: $821.93
- New price (at 5% yield): $783.53
- Loss: 4.67% (matches duration of ~4.67 years)
5-year 4% coupon bond:
- Original price: $1,000 (at par)
- New price (at 5% yield): $957.30
- Loss: 4.27% (duration ~4.27 years)
5-year 8% coupon bond:
- Original price: $1,169.86
- New price (at 5% yield): $1,125.39
- Loss: 3.79% (duration ~3.79 years)
All three mature in 5 years, but the zero-coupon bond loses the most in percentage terms (because its entire cash flow is far away, fully exposed to the rate change). The high-coupon bond loses less (because interim coupons arrive soon, reducing duration).
Next
Zero-coupon bonds represent one end of the spectrum: maximum duration for a given maturity. Coupon bonds fall between zero-coupon bonds and short-maturity bonds, with duration always less than maturity. The next article explores how coupon bonds' durations compare to their maturities and why.