Spot Curves vs. Par Curves

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title: "Spot Curves vs. Par Curves" description: "Master the difference between spot rates and par yields. Learn bootstrapping mechanics, pricing implications, and why using the wrong curve creates valuation errors." slug: "spot-curves-vs-par-curves" category: "Fixed Income" subcategory: "Yield Duration and Convexity" difficulty: "intermediate" readingTime: "7 min" author: "Equicurious" lastUpdated: "2025-12-29"

Using spot rates and par yields interchangeably is a pricing error waiting to happen. The valuation mistakes compound for longer maturities (where even small rate differences accumulate across multiple discounting periods). The Federal Reserve fits Treasury curves daily using the Svensson model with 6 parameters since 1980, and before that the Nelson-Siegel model when fewer securities existed (Gurkaynak, Sack, and Wright, 2006). The practical point: knowing which curve to use, and when, separates accurate bond pricing from rough approximations.

What Each Curve Actually Represents (The Core Distinction)

Par curve shows the coupon rate that would make a bond price at exactly par ($100) for each maturity. It answers: "What coupon rate prices a bond at face value given today's rates?" Treasury par yields are derived from closing market bid prices from the Federal Reserve Bank of New York at approximately 3:30 PM each business day.

Spot curve (also called the zero-coupon curve) shows the yield on a hypothetical zero-coupon bond for each maturity. It answers: "What's the correct discount rate for a single cash flow occurring at time T?" Spot rates discount each cash flow individually, without any reinvestment assumption baked in.

The point is: Par yields blend rates across maturities (because coupon bonds have multiple cash flows). Spot rates isolate the rate for a specific maturity point. When the yield curve has any slope, these two curves diverge.

Why the Difference Matters for Pricing

A coupon bond pays cash flows at multiple points in time. Each cash flow should be discounted at the rate appropriate for when it arrives (not at some blended average rate).

Using par yields to discount cash flows: You're applying a single blended rate to all cash flows. This creates systematic errors. For a 10-year bond, you're discounting the Year 1 coupon at a rate that includes information about 10-year money (which is wrong because Year 1 cash flows arrive in Year 1).

Using spot rates to discount cash flows: Each cash flow gets its own appropriate discount rate. The Year 1 coupon is discounted at the 1-year spot rate. The Year 5 coupon at the 5-year spot rate. The final principal at the 10-year spot rate. This produces the correct price.

The durable lesson: Spot rates are the building blocks of fixed income valuation. Par yields are derived statistics useful for quick comparisons and quoting conventions, but they're not the fundamental discount rates.

Bootstrapping: How Spot Rates Are Derived From Par Yields

You can observe par yields from market prices, but spot rates must be calculated. The bootstrapping method works by forward substitution, solving for each spot rate sequentially from the shortest maturity outward (Analyst Prep, 2024).

Step-by-Step Bootstrapping Example

Setup: You observe three par yields in the market:

• 1-year par yield: 4.00%
• 2-year par yield: 4.50%
• 3-year par yield: 5.00%

Step 1: Find the 1-year spot rate

A 1-year par bond has only one cash flow (coupon plus principal). So the 1-year spot rate equals the 1-year par yield.

S1 = 4.00%

Step 2: Find the 2-year spot rate

A 2-year par bond pays: $4.50 at Year 1, $104.50 at Year 2 (coupon plus face value). It prices at $100.

$100 = $4.50/(1 + S1) + $104.50/(1 + S2)^2

We know S1 = 4.00%, so:

$100 = $4.50/1.04 + $104.50/(1 + S2)^2

$100 = $4.327 + $104.50/(1 + S2)^2

$95.673 = $104.50/(1 + S2)^2

(1 + S2)^2 = 1.0923

S2 = 4.51%

Step 3: Find the 3-year spot rate

A 3-year par bond pays: $5.00 at Year 1, $5.00 at Year 2, $105.00 at Year 3. It prices at $100.

$100 = $5.00/(1.04) + $5.00/(1.0451)^2 + $105.00/(1 + S3)^3

$100 = $4.808 + $4.577 + $105.00/(1 + S3)^3

$90.615 = $105.00/(1 + S3)^3

S3 = 5.03%

Interpretation: Notice how spot rates are slightly higher than par yields when the curve slopes upward. This is because par yields blend in lower short-term rates through the coupon payments.

The Shape Relationship: When Curves Diverge Most

Steep upward-sloping curve: Spot rates exceed par rates at longer maturities. The gap widens as maturity increases. Why? Par yields are dragged down by the lower short-term rates embedded in coupon discounting.

Flat curve: Spot and par curves converge. When all maturities have the same rate, blending doesn't distort anything.

Inverted curve: Spot rates fall below par rates at longer maturities. The short-term coupon discounting at higher rates pulls par yields above pure long-term spot rates.

Current Market Context: December 2025 Treasury Yields

As of December 2025, the Treasury curve shows (Source: Federal Reserve H.15):

The 2-year to 10-year spread is approximately +8 bps (slightly positive, nearly flat). This near-flat environment means spot and par curves are relatively close. But the 30-year at 4.80% creates meaningful divergence at the long end. Spot rates for 25-30 year maturities would be higher than stated par yields because the long-term par yield gets dragged down by discounting intermediate coupons at lower mid-curve rates.

Practical Applications: Which Curve for Which Task?

Use par yields when:

• Quoting comparable yields to market participants (the convention)
• Comparing on-the-run Treasury issues (they trade near par)
• Quick screening and relative value discussions

Use spot rates when:

• Pricing individual bonds accurately (discount each cash flow correctly)
• Deriving forward rates (forward rates come from spot rates, not par yields)
• Building interest rate models (spot rates are the fundamental inputs)
• Valuing complex securities with uneven cash flows

Detection Signals: You're Likely Using the Wrong Curve If...

• You price a 30-year bond by discounting all cash flows at the 30-year par yield (instead of using term-appropriate spot rates)
• Your forward rate calculations produce illogical values (you probably started from par rates instead of spots)
• Your valuation of a zero-coupon bond differs significantly from market prices (zeros should price exactly using spot rates)
• You're comparing yields across bonds with different coupon structures without adjusting for the spot/par difference

Common Mistakes and How to Avoid Them

Mistake 1: Treating par yield as a discount rate

Par yield is a summary statistic describing what coupon makes a bond price at 100. It's not a discount rate for arbitrary cash flows.

Mistake 2: Ignoring the bootstrap when building a curve

Off-the-run securities provide more accurate curve fitting due to lower liquidity premiums versus on-the-run issues (Gurkaynak, Sack, and Wright, 2006). Bootstrapping from carefully selected securities matters.

Mistake 3: Forgetting curve shape effects

In steep curve environments, pricing a 10-year bond using the 10-year par yield versus properly bootstrapped spot rates can produce errors of 10-30 bps in implied yield. For a bond with duration of 8, that's 0.8-2.4% in price error.

Checklist: Spot vs. Par Curve Usage

Essential (get these right first)

• Price coupon bonds using spot rates for each individual cash flow
• Bootstrap spot rates from observable par yields using the sequential method
• Use par yields for quoting conventions and quick comparisons only

High-impact refinements

• Check curve construction methodology for institutional data feeds (Svensson vs. Nelson-Siegel parameters)
• Verify that your forward rate derivations start from spot rates
• Reconcile zero-coupon bond prices against your spot curve to validate accuracy

Practical Exercise: Build Your Own 5-Year Spot Curve

Take current Treasury par yields for 1, 2, 3, 4, and 5-year maturities. Apply the bootstrap method step by step. Compare your derived spot rates to the par yields. The divergence shows you exactly how much the par curve understates long-term spot rates in the current environment. This exercise takes 15 minutes with a calculator and reveals how spot/par confusion creates pricing errors.

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Related: Nominal Yield, Current Yield, and Yield to Maturity | Understanding Treasury Yield Curve Shapes | Forward Rate Derivation from the Curve