Bootstrapping the Spot Rate Curve: A Worked Example

Bootstrapping the spot rate curve is the foundational arithmetic of fixed-income pricing. By working backward from a sequence of par-priced coupon bonds, traders and analysts extract the zero-coupon spot rates—the discount rates that price each maturity individually. This method reveals the true shape of the yield curve and enables accurate bond valuation and interest rate risk assessment.

Why bootstrapping matters

In real markets, Treasury bonds and corporate bonds always pay coupons—they don’t come as pure zero-coupon instruments. Yet traders and risk managers need zero-coupon spot rates to properly discount future cash flows and compare bonds of different maturities.

Bootstrapping solves this problem. It uses the price of a coupon bond—observable in the market—to back out the implied zero-coupon rate for that maturity. By starting with the shortest-maturity bond and working forward, you build an entire spot curve.

This curve is essential because yield-to-maturity (YTM) blurs maturities together: a bond’s YTM is a single rate that discounts all its cash flows equally. But the 2-year rate and the 10-year rate should never be the same. Bootstrapping separates them.

A worked example: three bonds

Suppose you have three par-priced Treasury bonds. “Par-priced” means the bond’s market price equals its face value (100). These bonds pay annual coupons.

Your goal: find the spot rates s₁, s₂, and s₃ for each maturity.

Step 1: Extract the 1-year spot rate

The 1-year bond pays one cash flow: face value plus final coupon.

• Cash flow in 1 year: $100 + (0.02 × $100) = $102

The bond price is:

100 = 102 / (1 + s₁)

Solving for s₁:

(1 + s₁) = 102 / 100 = 1.02

s₁ = 0.02 = 2.0%

The 1-year spot rate is 2%. This makes sense: if the bond pays 2% coupon and trades at par, the zero-coupon rate must also be 2%.

Step 2: Extract the 2-year spot rate

The 2-year bond pays two cash flows:

• Year 1: $3 (coupon)
• Year 2: $100 + $3 = $103 (final coupon plus principal)

The bond price equation is:

100 = 3 / (1 + s₁) + 103 / (1 + s₂)²

You already know s₁ = 0.02. Substituting:

100 = 3 / 1.02 + 103 / (1 + s₂)²

100 = 2.94 + 103 / (1 + s₂)²

97.06 = 103 / (1 + s₂)²

(1 + s₂)² = 103 / 97.06 = 1.0612

(1 + s₂) = √1.0612 = 1.0301

s₂ = 0.0301 = 3.01%

The 2-year spot rate is approximately 3.01%. Notice it’s higher than the 1-year rate—the curve is upward sloping.

Step 3: Extract the 3-year spot rate

The 3-year bond pays three cash flows:

• Year 1: $4
• Year 2: $4
• Year 3: $100 + $4 = $104

The price equation is:

100 = 4 / (1 + s₁) + 4 / (1 + s₂)² + 104 / (1 + s₃)³

Substitute known values:

100 = 4 / 1.02 + 4 / (1.0301)² + 104 / (1 + s₃)³

100 = 3.92 + 3.77 + 104 / (1 + s₃)³

100 = 7.69 + 104 / (1 + s₃)³

92.31 = 104 / (1 + s₃)³

(1 + s₃)³ = 104 / 92.31 = 1.1263

(1 + s₃) = ∛1.1263 = 1.0405

s₃ = 0.0405 = 4.05%

The 3-year spot rate is approximately 4.05%.

Your bootstrapped curve

This upward-sloping curve tells you the market is pricing higher yields for longer-dated cash flows—a normal environment. You can now use these rates to value any other bond, strip its duration, or calculate interest rate risk.

Why the order matters

Bootstrapping must proceed from shortest to longest maturity. The 1-year bond has only one cash flow, so solving for s₁ is trivial. The 2-year bond has two cash flows, but you’ve already pinned down s₁, leaving only s₂ as an unknown. The 3-year bond has three cash flows, but you’ve already solved for s₁ and s₂, leaving only s₃.

If you tried to solve for the 3-year rate before the 2-year rate, you’d have two unknowns in one equation—impossible. The sequential structure of bootstrapping is not arbitrary; it’s the only way forward.

Practical notes

In real markets, you rarely have exactly par-priced bonds at convenient annual intervals. Practitioners interpolate between observed bond prices and use more sophisticated techniques—like spline methods—to smooth the curve. SOFR and LIBOR swap curves are bootstrapped from futures and swap contracts, not just bonds.

Bootstrapping also assumes no arbitrage—that the spot rates you extract are consistent with market prices and that no two bonds offer free profit. In efficient markets, this holds; in stressed conditions, it may not.

See also