Par Curve vs Zero Curve
Par Curve vs Zero Curve
Two different but mathematically equivalent representations of the Treasury term structure exist: the par curve and the zero curve. The par curve shows the yield to maturity of a hypothetical Treasury bond trading at par (100% of face value) at each maturity. The zero curve shows the spot rate (the zero-coupon rate) at each maturity. Both curves contain identical information about the market's valuation of bonds at different maturities, but they present that information in different ways.
The par curve is what you see in the financial media: "The 10-year Treasury yields 4.2%." This is the par yield—the coupon rate that would make a 10-year Treasury bond issued today trade at exactly par. The zero curve, by contrast, shows the discount rates: "The 10-year spot rate is 4.25%." These are two ways of describing the same market reality, but practitioners sometimes prefer one or the other depending on the use case.
Understanding both representations helps you interpret Treasury yields correctly and work with bond data from different sources. Some bond databases report par curves; others report zero curves. Some traders work primarily with zero curves for valuation; others prefer par curves for intuitive understanding. Most modern financial systems display both.
Key takeaways
- The par curve shows coupon yields of hypothetical par bonds at each maturity; the zero curve shows zero-coupon spot rates.
- The two curves are mathematically equivalent; you can convert one to the other using standard formulas.
- Par curves are more intuitive to the average investor; hearing "the 10-year Treasury yields 4.2%" is instantly understandable.
- Zero curves are more theoretically pure and are preferred for bond valuation, because they directly discount cash flows at each maturity.
- In upward-sloping environments, zero rates are typically higher than par yields for long maturities; in inverted environments, the relationship reverses.
Defining Par Curve and Zero Curve
The par curve is constructed as follows: for each maturity n, find the coupon rate c such that a bond with coupon rate c, priced to yield to maturity c (i.e., discounted at rate c), has a price of exactly 100 (par). This coupon rate is the par yield for that maturity. By definition, all par bonds trade at 100; they have been constructed to do so.
The zero curve is the spot rate curve: the discount rate at each maturity n such that $1 received n years from now has a present value implied by the Treasury yield curve.
In a simple case, suppose all spot rates are 3% (a flat zero curve). The par curve would also be flat at 3%, because a bond with 3% coupon discounted at 3% would trade at par. But suppose spot rates rise from 2% at 1 year to 4% at 10 years (an upward-sloping zero curve). The par curve at the 10-year maturity would be different from 4%, because the 10-year par rate is a weighted average of the expected path of returns over 10 years, not just the 10-year spot rate.
Mathematical Relationship Between Par and Zero Curves
If you know the zero curve, you can construct the par curve. The price of a par bond is given by:
100 = (c / 2) / (1 + s1/2) + (c / 2) / (1 + s2/2)^2 + ... + (100 + c / 2) / (1 + s10/2)^10
where c is the semi-annual coupon (half the annual par rate), and s1, s2, ..., s10 are the semi-annual spot rates.
Solving for c gives the par rate at each maturity. Conversely, if you know the par curve, you can bootstrap backward to find the zero curve, as described earlier.
In practice, financial institutions compute both curves and publish them together. The curves will be similar in shape (both upward-sloping, flat, or inverted) but may differ in level, especially at long maturities.
When Par and Zero Curves Differ Significantly
In a normal upward-sloping environment, where spot rates rise as maturity extends, the par curve will typically be lower than the zero curve at long maturities. A 30-year spot rate might be 4.2%, but the 30-year par rate (the coupon needed to make a 30-year bond trade at par) might be 4.1%. The difference reflects the fact that coupon-bearing bonds benefit from reinvestment of intermediate coupons at rates higher than the spot rate at that point on the curve, which offsets some of the duration risk.
In an inverted environment, where spot rates fall as maturity extends, the relationship can reverse. The par curve might trade above the zero curve at long maturities, because bondholders are foregoing the reinvestment benefit at lower rates.
The magnitude of the par-zero difference depends on the slope of the curve and the convexity of the cash flow structure. A very steep curve (e.g., 2-year spot 2%, 30-year spot 5%) will show large par-zero differences at the long end. A nearly flat curve will show small differences.
Practical Implications for Bond Investors
For individual investors and passive bond fund holders, understanding the par-zero distinction is less critical, because the impact on returns is already embedded in bond prices. A bond fund's performance reflects the entire term structure, whether it is represented as a par curve or a zero curve.
But for bond traders and portfolio managers building tactical positions, the choice between par and zero curves matters. If you are comparing the relative value of bonds at different maturities, the zero curve is the more direct tool, because it shows you the pure discount rate at each maturity. If you are trying to understand what coupons future Treasury issues might have, the par curve is more relevant.
A practical example: suppose you want to determine if the 5-year Treasury is cheap or expensive relative to the 10-year. You might look at the par curve and see that 5-years yield 3.5% and 10-years yield 4%. The 35 basis point spread might seem modest. But if you look at the zero curve, you might see that 5-year spot rate is 3.6% and 10-year spot rate is 4.2%. The 60 basis point spread in spot rates tells a different story—the long end is more compensated for its extra risk. This could influence your decision about whether to extend duration.
How Central Banks and Statisticians Use the Curves
Central banks like the Federal Reserve publish both par and zero curves. The Fed's yield curve is published as the par curve (coupon yields), but the underlying data is derived from a combination of par and zero concepts. The Fed uses a smoothing method based on Svensson-Nelson methodology, which fits both par and zero curves simultaneously and ensures they are consistent with each other.
Academic researchers and econometricians often prefer the zero curve because it is theoretically pure and removes coupon effects. When studying the relationship between the yield curve and economic growth, researchers use zero curves to avoid having their results contaminated by coupon artifacts.
Credit analysts and corporate bond traders use zero curves plus credit spreads to value corporate bonds. A corporate bond's price is the sum of its cash flows discounted at Treasury zero rates plus a credit spread. The Treasury par curve alone is not sufficient for this purpose; you need the zero curve to discount cash flows precisely.
Flowchart: Choosing Between Par and Zero Curves
Historical Example: Par vs Zero in 2022
In 2022, as the Fed raised rates aggressively from 0% to 4.25%, the Treasury curve underwent dramatic shifts. The par curve inverted (2-year yielding more than 10-year). The zero curve also inverted, but the inversion was more pronounced. At some points, the 10-year zero rate was 100+ basis points below the 2-year zero rate, while the par curve showed a smaller inversion.
Traders who relied solely on the par curve (as published in the financial media) might have underestimated how much the term structure had inverted. The zero curve, with its larger inversion, better reflected the market's extreme skepticism about long-term growth and rates.
This episode shows that in very unusual market conditions (sharp trend changes, high volatility), the par and zero curves can diverge visibly. Practitioners who understand both representations are better equipped to interpret what the market is pricing.
Data Sources and Curve Selection
If you are working with bond data, you will encounter both par and zero curves depending on your source. Bloomberg terminals display both. The Federal Reserve publishes its official par curve (the Gurkaynak-Sack-Swanson model par curve). The Treasury Department's website shows daily par curve data. Academic databases often include zero curves because researchers prefer them.
For individual investors using online brokerage accounts or financial websites, the curves shown are par curves. This is fine for everyday use. If you are building a bond portfolio or doing more sophisticated analysis, you may want to access zero curve data through Bloomberg, specialized bond analytics platforms, or academic sources.
The good news is that par and zero curves are mathematically equivalent, so you can convert between them if you have one and need the other. Most financial software provides both automatically. The choice between them is mainly about convenience and convention.
Next
Par and zero curves are two ways of describing the same term structure. But why does the curve slope the way it does? What economic factors determine the term premium—the extra yield demanded for long-term bonds? Two competing theories provide answers: pure expectations theory and liquidity preference theory.