Bootstrapping the Spot Rate Curve

Introduction

In our previous lessons, we've established the theoretical superiority of using a term structure of spot rates over a single yield-to-maturity for accurately pricing bonds. We even developed a Python function to derive a forward rate curve, assuming a complete spot curve was at our disposal. This naturally leads to a crucial practical question: how do we obtain this spot rate curve in the first place?

The market doesn't provide a neat table of spot rates. Instead, it gives us the prices of actively traded, coupon-bearing bonds. This lesson introduces bootstrapping, the fundamental iterative methodology used by financial professionals to construct a zero-coupon yield curve from the market prices of these coupon bonds. By the end of this lesson, you will be able to explain the entire bootstrapping process, from its underlying motivation to its core algorithmic steps.

1. The Motivation for Bootstrapping

Before diving into the mechanics, let's solidify why this process is necessary. As we've discussed, a bond's yield-to-maturity (YTM) is a single average rate that is mathematically convenient but theoretically flawed because it assumes a flat yield curve and a constant reinvestment rate for all coupons.

A more accurate valuation method involves treating a coupon bond as a portfolio of individual zero-coupon bonds and discounting each cash flow at a unique spot rate $s_{t}$ that corresponds to its specific maturity $t$ . The challenge is that the market for pure zero-coupon bonds (like US Treasury STRIPS) is not liquid across all maturities. We must, therefore, infer the spot rates from the instruments we can observe: coupon-bearing bonds.

The following video provides an excellent intuitive explanation of this problem, distinguishing between different yield curves and setting the stage for why bootstrapping is the required technique.

ep11: Yield curves - par curves, spot curves, bootstrapping...simple explanation

The video "ep11: Yield curves - par curves, spot curves, bootstrapping" by Matt Thomas offers a clear conceptual walkthrough of why spot rates are superior to YTM and introduces bootstrapping as the solution.

Please watch two segments from this video: First, focus on the discussion on the limitations of YTM. The speaker clearly explains why using a single discount rate for cash flows occurring at different times is unrealistic. Next, watch the segment that introduces bootstrapping as the process of converting an observable yield curve (like a YTM curve) into the more theoretically sound spot curve.

The video makes it clear: our goal is to build a spot curve to properly value any set of cash flows. Bootstrapping is the algorithm that gets us there using the prices of existing coupon bonds.

2. The Bootstrapping Algorithm

Bootstrapping is an iterative, sequential process. The name itself is an analogy for "pulling yourself up by your own bootstraps," as each step relies on the successful completion of the previous ones.

Let's assume we have a set of bonds of the same credit risk (e.g., government bonds) with a range of maturities (e.g., 6 months, 1 year, 1.5 years, etc.) and their current market prices. We will extract the spot rates one by one, from shortest to longest maturity. For simplicity in our formulas, let's assume semi-annual coupon payments and compounding, which is a common convention.

Step 1: The Anchor Point

We begin with the shortest-maturity instrument, ideally one with a single cash flow, like a 6-month Treasury bill. For a bond with only one payment, its yield is, by definition, the spot rate for that maturity.

Let $P_{0.5}$ be the price of a 6-month zero-coupon bond with a face value $F V$ . The relationship is:

P_{0.5} = \frac{F V}{( 1 + s _{0.5} /2 ) ^{2 \times 0.5}} = \frac{F V}{1 + s _{0.5} /2}

Since $P_{0.5}$ and $F V$ are known, we can solve directly for the 6-month annualized spot rate, $s_{0.5}$ :

s_{0.5} = 2 \times (\frac{F V}{P _{0.5}} - 1)

This first spot rate is our "anchor" for the entire process.

Step 2: The First Iteration

Next, we take the 1-year bond. Assume it pays a semi-annual coupon $C$ . Its market price, $P_{1}$ , must equal the present value of its two cash flows, each discounted by the correct spot rate.

The first coupon at 6 months is discounted by $s_{0.5}$ .
The final cash flow (coupon + face value) at 1 year is discounted by $s_{1}$ .

The pricing equation is:

P_{1} = \frac{C /2}{( 1 + s _{0.5} /2 ) ^{1}} + \frac{F V + C /2}{( 1 + s _{1} /2 ) ^{2}}

In this equation, $P_{1}$ , $C$ , and $F V$ are known characteristics of the bond. Crucially, $s_{0.5}$ is also known from our previous step. The only unknown variable is the 1-year spot rate, $s_{1}$ . We can now algebraically rearrange the equation to solve for $s_{1}$ .

The following video segment demonstrates this exact iterative step, solving for the 2-year spot rate using the known 1-year rate. The logic is identical to our step-by-step process.

Bootstrapping Spot Rates From the Par Curve

In his video "Bootstrapping Spot Rates From the Par Curve," Ryan O'Connell provides a very clear, step-by-step algebraic derivation. We will focus on just one of these steps to understand the core iterative logic.

Watch the derivation of the two-year spot rate. Notice how the price of the 2-year bond (assumed to be at par, 100) is set equal to its cash flows. The first cash flow is discounted using the known 1-year spot rate, leaving the 2-year spot rate as the only unknown to solve for.

Step 3: The General Case

The process continues sequentially. To find the spot rate for maturity $T_{n}$ , we use a bond that matures at $T_{n}$ . Its price, $P_{n}$ , is given by:

P_{n} = i = 1 \sum n - 1 \frac{C _{i}}{( 1 + s _{T_{i}} ) ^{T_{i}}} + \frac{F V + C _{n}}{( 1 + s _{T_{n}} ) ^{T_{n}}}

Note: For simplicity, the equation is shown with annual compounding. You would adjust the exponents and rates for semi-annual or other frequencies.

At each step $n$ , the spot rates $s_{T_{1}}, s_{T_{2}}, \dots, s_{T_{n - 1}}$ have already been calculated. Therefore, we can compute the present value of all intermediate cash flows. Subtracting this sum from the bond's price $P_{n}$ leaves us with the present value of the final cash flow. From there, we can solve for the unknown spot rate $s_{T_{n}}$ .

This recursive dependency is the essence of the bootstrapping algorithm.

3. Data Requirements and Practical Hurdles

The elegant theory of bootstrapping relies on a specific set of inputs that can be challenging to find in the real world.

A "Chain" of Bonds: The method requires a set of bonds with maturities spaced out at regular intervals (e.g., 0.5, 1.0, 1.5, 2.0 years, etc.). If there is a gap—for example, no liquid bond exists with a 1.5-year maturity—we cannot directly compute the 1.5-year spot rate.
Liquidity Issues: The most recently issued government bonds, known as "on-the-run" securities, are typically very liquid. However, there are not enough of them to build a full yield curve. Analysts must therefore use older, "off-the-run" bonds to fill in the maturity gaps. These bonds may be less liquid, and their prices may contain more "noise" (e.g., wider bid-ask spreads).
Overfitting: A significant drawback of this method is that it perfectly fits the observed bond prices. If a bond's price has an error or is an outlier due to idiosyncratic factors (like tax effects or low liquidity), bootstrapping will absorb this error directly into the calculated spot rate. This can result in a jagged or non-smooth spot curve, whereas we generally expect the term structure to be smooth.

The following article delves into the practicalities of collecting bond data and running the bootstrapping process, highlighting the need to combine on-the-run and off-the-run bonds to get a usable dataset.

Computing Risk Free Rates and Excess Returns Part 2

This article from LSEG provides a practical guide with Python code for bootstrapping. We will focus on its discussion of the data challenges.

Please read the section titled "Example with US Treasury Bonds with Fixed Coupon Rates and using Semi-Annual Compounding." Pay special attention to the subsections "On-the-run sovereign Bonds" and "On and Off-the-run sovereign Bonds." Notice how the author first shows the scarcity of on-the-run bonds and then demonstrates the necessity of including off-the-run bonds to assemble a workable set of instruments for bootstrapping.

Conclusion

In this lesson, we have demystified the process of bootstrapping. You now understand that it is not a complex black box, but a logical, iterative algorithm for uncovering the hidden term structure of spot rates from observable bond prices.

Key Takeaways:

Purpose: Bootstrapping is necessary because a full range of liquid zero-coupon bonds does not exist. It allows us to derive the theoretically correct spot rates from available coupon-bond prices.
Methodology: It is a sequential algorithm that starts with the shortest maturity bond and iteratively solves for the next spot rate using the rates already determined.
Core Equation: At each step, the bond's market price is set equal to the sum of its discounted cash flows. All but one of the required spot rates are known from previous steps, allowing the final unknown spot rate to be solved for algebraically.
Practicality: The method requires a carefully selected set of bonds to cover the desired maturity spectrum, and it is sensitive to noise and errors in input bond prices.

In our next session, we will formalize this understanding by performing a complete, step-by-step derivation of a multi-year spot curve starting from a "par-yield curve"—a common theoretical starting point that simplifies the initial calculations.

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