Spot Rates and Bond Pricing with ZCYC

Introduction

Welcome to the second module of our course. In the previous lesson, we concluded our exploration of Yield-to-Maturity (YTM) by highlighting its theoretical limitations, namely the unrealistic reinvestment assumption and the "coupon effect." We established that a more accurate valuation method involves using a different discount rate for each cash flow, based on its timing.

This lesson marks our formal entry into the world of term structure models. We will define the foundational concepts of spot rates and the zero-coupon yield curve (ZCYC). You will learn the theoretically sound method for pricing a bond by decomposing it into a series of zero-coupon instruments and discounting each using the appropriate spot rate. This will provide the "why" and "how" behind the pricing formula, setting the stage for its Python implementation in our next session.

1. From Zero-Coupon Bonds to Spot Rates

To build a robust pricing model, we must start with the most basic fixed-income security: the zero-coupon bond. As its name suggests, this is a bond that pays no coupons. It is bought at a discount to its face value and pays its full face value at maturity.

To formally define this concept and link it to spot rates, please review the following text.

Term Structure of Interest Rates

This handout from a Financial Mathematics course at Millersville University provides concise, formal definitions of zero-coupon bonds, the term structure, and spot rates.

Read the section "Zero Coupon Bonds" to understand their structure. Then, read the section "Term Structure of Interest Rates" to get the formal definition of a spot rate (s_t) and its relationship to the price of a zero-coupon bond.

As you've just read, the spot rate, denoted $s_{t}$ , is the annual effective yield rate for a single cash flow to be received at a specific point in the future, time $t$ . In other words, it is the yield-to-maturity of a zero-coupon bond maturing at time $t$ .

The price $P$ of a zero-coupon bond maturing in $t$ years with a face value of $F V$ is therefore:

P = \frac{F V}{( 1 + s _{t} ) ^{t}}

Each spot rate is "pure" in the sense that it reflects the time value of money for a single, specific maturity, with no confounding effects from coupon reinvestments.

2. The Zero-Coupon Yield Curve (ZCYC)

The Zero-Coupon Yield Curve (ZCYC), also known as the spot curve or zero curve, is simply the set of all spot rates for a range of maturities at a single point in time. It is a graphical representation of the term structure of interest rates.

Par Coupon And Zero Rate Curves — This graph illustrates a typical upward-sloping Zero Rate curve (in red), which is the ZCYC. It shows the unique yield for zero-coupon bonds of varying maturities. For comparison, it also shows a Par Coupon Yield curve, which we will discuss in a later module.

This curve is the fundamental tool for valuation. While the YTM of a 10-year coupon bond is a complex average, the 10-year spot rate from the ZCYC has an unambiguous meaning: it is the rate of return for locking money away for exactly 10 years in a zero-coupon instrument.

3. Pricing a Coupon Bond using the ZCYC

With the concept of the ZCYC established, we can now derive the correct pricing formula for a standard coupon-bearing bond.

The key insight is to view a coupon bond not as a single instrument, but as a portfolio of zero-coupon bonds. Each coupon payment and the final principal repayment can be treated as an individual zero-coupon bond maturing on its respective payment date.

To find the bond's price, we simply calculate the present value of each individual cash flow using the spot rate that corresponds to its specific maturity, and then sum these present values.

The price $P$ of a bond with $n$ cash flows $C F_{1}, C F_{2}, \dots, C F_{n}$ occurring at times $T_{1}, T_{2}, \dots, T_{n}$ is given by:

P = \frac{C F _{1}}{( 1 + s _{T_{1}} ) ^{T_{1}}} + \frac{C F _{2}}{( 1 + s _{T_{2}} ) ^{T_{2}}} + \dots + \frac{C F _{n}}{( 1 + s _{T_{n}} ) ^{T_{n}}}

Which can be written more compactly as:

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

where $s_{T_{i}}$ is the spot rate for the maturity $T_{i}$ .

This pricing method is free from the flaws of YTM. It makes no reinvestment assumption and is consistent with the no-arbitrage principle, as the bond's value is simply the sum of its constituent parts, priced according to the prevailing term structure.

Let's look at a worked example.

Term Structure of Interest Rates

This section of the handout provides a clear, step-by-step example of this pricing methodology.

Focus on the solution for the "4% annual coupon bond" and the "9% annual coupon bond". Note how the provided spot rates are semi-annual, so the formula is adjusted for the compounding frequency m=2: the rate is divided by 2, and the exponent is the number of periods (years x 2). This demonstrates the general formula: P = \sum_{i=1}^{n} \frac{CF_i}{\left(1+\frac{s_{T_i}}{m}\right)^{m \cdot T_i}}

To see another application of this principle, watch the following video segment. It uses a set of pre-determined spot rates to price a bond.

Bonds: Spot Rates from Forward Rates

This video from Arnold Tutoring provides a dynamic walkthrough of pricing a bond using a set of spot rates.

Focus only on the final part of the video, which calculates the bond's price. Observe how each of the three cash flows—the two coupons and the final principal plus coupon—is discounted using its own distinct spot rate corresponding to its maturity (1, 2, and 3 years).

4. Continuous Compounding and Discount Factors

For many quantitative applications, it is mathematically more convenient to work with continuously compounded rates and discount factors. Given your background, this formulation will likely feel familiar and is essential for more advanced models.

A discount factor, $D F (T)$ , is the price today of receiving $1 at time $T$ . If we use a continuously compounded spot rate, $s_{T}^{c}$ , the relationship is:

D F (T) = e^{- s_{T}^{c} \cdot T}

The bond pricing formula then becomes an elegant sum of discounted cash flows:

P = i = 1 \sum n C F_{i} \cdot D F (T_{i}) = i = 1 \sum n C F_{i} \cdot e^{- s_{T_{i}}^{c} \cdot T_{i}}

This approach simplifies many derivations, especially when dealing with derivatives of the price function, as we will see when we study duration and convexity.

The following article, while focused on bootstrapping (a topic for our next module), provides the key formula for calculating discount factors from a zero-coupon yield curve.

Python Bootstrapping the Zero Curve

This LinkedIn article discusses the practical steps of building a yield curve. We are interested in its final conclusion about using the constructed curve.

Read only the final two paragraphs of the article, starting from "Once we have the zero coupon yield curve...". This section explicitly states the formula for calculating a discount factor from a continuously compounded zero rate (spot rate).

Conclusion

In this lesson, we established the theoretical foundation for modern bond pricing. We moved away from the single-rate YTM model and embraced the term structure of interest rates.

Key Takeaways:

A spot rate ( $s_{t}$ ) is the yield on a zero-coupon bond maturing at time $t$ , representing the pure time value of money for that specific maturity.
The Zero-Coupon Yield Curve (ZCYC) is the collection of spot rates across the maturity spectrum and serves as the fundamental tool for valuation.
The no-arbitrage price of a coupon bond is the sum of its cash flows, where each cash flow is discounted by the spot rate corresponding to its payment date.
The pricing formula can be elegantly expressed using discount factors and continuously compounded rates, a formulation that is common in quantitative finance.

You now have the complete theoretical framework for pricing a bond using a yield curve. In the next lesson, we will put this theory into practice by implementing a Python function to price a coupon-bearing bond by discounting each cash flow with its corresponding spot rate.

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