YTM vs. Term Structure: Pricing & Limitations

Introduction

In our previous lessons, we built a solid foundation for bond valuation, culminating in solving for a bond's Yield-to-Maturity (YTM) from its market price. We treated YTM as the single, constant discount rate that makes the present value of all future cash flows equal to the bond's price. Today, we will critically evaluate this single-rate assumption.

This lesson directly addresses a core subtlety in fixed income: the distinction between pricing with a single YTM versus a term structure of interest rates. We will explore the theoretical limitations of YTM, setting the stage for more sophisticated and accurate pricing models that you will build in the upcoming modules. By the end of this lesson, you will understand why the concept of a yield curve is fundamental to modern bond analytics.

1. Two Approaches to Discounting

So far, our pricing formula has been:

P = i = 1 \sum n \frac{C F _{i}}{( 1 + y / f ) ^{t_{i}}}

where $y$ is the single YTM, and $t_{i}$ is the $i$ -th period. The key assumption is that one rate, $y$ , applies to all cash flows, regardless of whether they are received in 6 months or 10 years. This implies a flat yield curve.

A more theoretically sound approach recognizes that the appropriate discount rate should depend on the timing of the cash flow. This leads to the concept of the term structure of interest rates, or the spot rate curve. A spot rate, $z_{t}$ , is the rate of interest for a risk-free investment held from today until a specific maturity date, $t$ . It is the yield on a zero-coupon bond maturing at time $t$ .

Using spot rates, the pricing formula becomes:

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

Here, each cash flow $C F_{i}$ occurring at time $T_{i}$ is discounted by its own unique spot rate, $z_{i}$ .

The following video provides an excellent illustration of these two competing pricing methodologies.

Yield to Maturity Interpretations (FRM T3-10)

This video from Bionic Turtle clearly contrasts pricing a bond using a spot rate curve versus using a single YTM.

Watch the section that prices a bond using a spot rate curve. Notice how each cash flow is discounted by a different rate. Then, watch the following segment which defines YTM as the single rate that forces the calculated price to match the observed price, visually representing it as a flat line.

2. The Theoretical Flaws of Yield-to-Maturity

The YTM is a convenient and widely quoted metric, but its simplifying assumption introduces several critical theoretical limitations.

a) The Reinvestment Assumption

The most significant flaw in YTM is the implicit assumption about reinvesting coupon payments. A bond's YTM will only be its actual, realized rate of return if two conditions are met:

The bond is held to maturity.
All intermediate coupon payments are reinvested at a rate equal to the original YTM.

In a world where interest rates fluctuate and the yield curve is not flat, the likelihood of reinvesting every coupon at the exact initial YTM is virtually zero. This is known as reinvestment risk.

The final segment of the Bionic Turtle video explains this crucial point.

Yield to Maturity Interpretations (FRM T3-10)

This final part of the video clarifies the conditions under which YTM equals the realized return.

Watch the section explaining why YTM is not the realized return. Focus on the reinvestment assumption, which is the key takeaway.

If future reinvestment rates are lower than the YTM, the realized return will be lower. Conversely, if they are higher, the realized return will be higher. Pricing with a spot rate curve does not suffer from this issue, as it makes no assumptions about reinvestment. It simply gives the present value of a known stream of cash flows based on the current term structure.

b) The "Coupon Effect": YTM as a Flawed Average

Mathematically, a bond's YTM can be seen as a complex, weighted average of the underlying spot rates that form the true term structure. This creates a problem known as the coupon effect. If the yield curve is not flat, two bonds with identical maturities but different coupon rates will have different YTMs.

A Review of Term Structure Estimation Methods

This paper from the CAIA Association provides a formal definition of YTM and its limitations.

Read section 1.3. Yield to Maturity. Pay close attention to the statements: "the yield to maturity is a complex weighted average of zero-coupon rates" and the explanation of the "coupon effect."

Why does this happen? A high-coupon bond pays a larger proportion of its total cash flow sooner than a low-coupon bond of the same maturity. Its value is therefore more heavily influenced by the short-term spot rates. Conversely, a low-coupon bond's value is dominated by the long-term spot rates. When the yield curve is upward-sloping, the YTM of a high-coupon bond will be lower than that of a low-coupon bond with the same maturity, because it's more heavily weighted by the lower, short-term spot rates.

This makes it difficult to compare the yields of different bonds directly. Is a 10-year bond with a 5% YTM a better value than a 10-year bond with a 5.2% YTM? Not necessarily. The difference might simply be due to their different coupon rates and the shape of the spot curve.

c) The Arbitrage Problem

The use of a single YTM can lead to mispricings relative to the individual components of a bond's cash flows. In markets where zero-coupon bonds (like U.S. Treasury STRIPS) are traded, it's possible to buy and sell individual cash flows. If a coupon bond's price (based on its YTM) deviates from the total price of its individual stripped cash flows (priced using spot rates), a risk-free arbitrage opportunity exists.

Consider the following scenario:

Arbitrage Opportunity from Bond Pricing Discrepancy — This diagram illustrates how a bond's price, when calculated using a single YTM, can differ from the sum of its parts when each part is valued using the appropriate spot rate from the STRIPS market. This difference creates a risk-free arbitrage profit.

In this example:

A 4-year, 10% coupon bond is priced at $106.62 using a single YTM of 8%.
The market for Treasury STRIPS allows you to sell each of the bond's cash flows individually at prices that reflect the spot rate curve. The sum of these individual prices is $106.94.
An arbitrageur could buy the bond for $106.62, "strip" it into its four coupon payments and principal, and immediately sell them in the STRIPS market for $106.94, locking in a risk-free profit of $0.32.

This demonstrates that the theoretically correct price of a bond is the sum of the present values of its cash flows, with each cash flow discounted at the spot rate corresponding to its maturity. A single YTM is a simplifying abstraction that can violate this principle.

3. A Practical Look: YTM vs. Spot Rate Curves

Let's see what this looks like with real data. The following resource demonstrates building a YTM curve and a spot rate curve from the same set of U.S. Treasury bonds. While it introduces a technique called "bootstrapping" which we will cover in detail in Module 3, the key for now is to observe the visual difference between the two curves.

Computing Risk Free Rates and Excess Returns Part 2

This article from LSEG provides a Python-based example of building yield curves from market data.

Navigate to the section "On and Off-the-run sovereign Bonds". Skim the text and code to understand the goal: gathering data for a set of bonds with different maturities on a single day. Examine the chart titled "Yield To Maturity of US Treasury Bonds...". Notice the jagged, "zigzagging" pattern. This is the coupon effect in action! Bonds with different coupons have different YTMs, even for similar maturities. Now, look at the final chart in the section, titled "On- and Off-the-run US Treasury Bonds...". This chart overlays the YTM curve with a derived spot rate curve. Observe how the spot rate curve is much smoother and provides a more consistent representation of the term structure.

The messy YTM curve is a direct result of the limitations we've discussed. It's a collection of individual, bond-specific internal rates of return. The spot rate curve, by contrast, represents the pure, underlying term structure of interest rates on that day. This is the curve we need for consistent and accurate pricing, and learning how to construct it will be the focus of our next several lessons.

Conclusion

In this lesson, we moved beyond the convenience of YTM to understand its fundamental weaknesses. You are now equipped to distinguish between these two critical concepts in bond valuation.

Key Takeaways:

YTM Pricing uses a single, constant discount rate for all of a bond's cash flows, implying a flat yield curve.
Term Structure (Spot Rate) Pricing uses a unique, maturity-specific discount rate for each cash flow, which is more theoretically sound.
The three main limitations of YTM are:
1. Reinvestment Risk: YTM is only the realized return if all coupons are reinvested at the same YTM, which is unrealistic.
2. The Coupon Effect: YTM is a weighted average of spot rates, causing bonds with the same maturity but different coupons to have different YTMs.
3. Arbitrage: Pricing by YTM can create inconsistencies with the prices of a bond's constituent zero-coupon cash flows, leading to arbitrage opportunities.

You now understand why we need to go beyond YTM. In our next lesson, we will begin our journey into building and using the term structure by formally defining spot rates and the zero-coupon yield curve (ZCYC), and using them to price coupon-bearing bonds in Python.

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