Limitations of Linear Duration for Large Yield Changes

Introduction

In our last lesson, we successfully implemented Python functions to calculate a bond's Macaulay and Modified Duration, establishing a practical method for quantifying its first-order interest rate risk. We defined Modified Duration as the first derivative of the price-yield function, which allows us to create a linear estimate of a bond's price change for a given change in yield.

This lesson scrutinizes the accuracy of that linear approximation. The true relationship between a bond's price and its yield is not linear, but curved. We will explore why and when the duration-only approximation breaks down, particularly for larger movements in interest rates. Understanding this limitation is the critical first step before we can improve our risk model. We'll connect this to the mathematical concept of a Taylor series expansion, a framework likely familiar from your engineering and data science background, to formalize why duration is only the beginning of the story.

The Non-Linear Reality of the Price-Yield Curve

Let's recall the fundamental bond pricing equation, where the price $P$ is the sum of discounted cash flows:

$P (y) = \sum_{i = 1}^{N} \frac{C F _{i}}{( 1 + y / m ) ^{t_{i} \cdot m}}$

Here, the price $P$ is a function of the yield $y$ . Because the yield term is in the denominator with a power, the relationship is inverse and non-linear. Specifically, the function $P (y)$ is convex.

The linear approximation we derived using Modified Duration, $\frac{Δ P}{P} \approx - D_{mod} \cdot Δ y$ , is nothing more than the equation of a straight line tangent to this curve at the initial yield.

This is best understood visually.

Price Approximation using Duration — This graph shows the actual curved price-yield relationship for a bond (blue curve) and the linear approximation based on duration (red tangent line). The error of the approximation is the vertical distance between the line and the curve.

As you can see, for very small changes in yield around the initial point $y^{*}$ , the tangent line is a very close approximation of the actual price curve. However, as the yield change ( $Δ y$ ) becomes larger, the line and the curve diverge significantly. This divergence is the approximation error.

Taylor Series: The Mathematical Framework

The formal tool for approximating a function like $P (y)$ is the Taylor series expansion. Expanding the price function $P (y)$ around the initial yield $y_{0}$ , we get:

$P (y_{0} + Δ y) = P (y_{0}) + \frac{d P}{d y}_{y_{0}} Δ y + \frac{1}{2} \frac{d ^{2} P}{d y ^{2}}_{y_{0}} (Δ y)^{2} + \dots$

Dividing by the initial price $P (y_{0})$ to get the percentage change gives:

$\frac{Δ P}{P} \approx (\frac{1}{P} \frac{d P}{d y}) Δ y + \frac{1}{2 P} \frac{d ^{2} P}{d y ^{2}} (Δ y)^{2} + \dots$

From our previous lesson, we know that Modified Duration is $D_{mod} = - \frac{1}{P} \frac{d P}{d y}$ . Substituting this into the expansion, we see that the duration-based approximation uses only the first-order term:

$\frac{Δ P}{P} \approx - D_{mod} \cdot Δ y$

This formally shows that our duration approximation is a first-order Taylor expansion, which is inherently a linear approximation. The document "Lecture 6 Fixed Income Portfolio Management" provides a good overview of this application.

Lecture 6: Using Taylor Series Expansions

This resource from Simon Fraser University introduces the use of Taylor series for approximating bond price changes.

As you read, focus on these key ideas: The section "Using Taylor Series Expansions" explains that simpler functions (like duration) are used to approximate the more complex bond price function. The section "Evaluating the Taylor Series" notes that the sum of the first two terms of the expansion gives the linear approximation. The section "Zero order, First order..." explicitly identifies the first-order approximation as a line with a slope equal to the first derivative—this is our duration approximation.

Visualizing and Quantifying the Error

The crucial insight from the graph and the Taylor series is that the duration approximation error is systematic. Because the true price-yield curve is convex (it curves "upwards"), the tangent line will always lie below the curve.

This leads to two specific inaccuracies:

For a decrease in yield: The actual price increase is larger than the increase predicted by duration. The approximation underestimates the new price.
For an increase in yield: The actual price decrease is smaller than the decrease predicted by duration. The approximation also underestimates the new price (by predicting a price that is too low).

In short, the duration approximation is always less optimistic than reality. Let's watch a video that provides a clear, practical demonstration of this effect using a spreadsheet.

Calculate Bond Convexity and Duration in Excel | Interest Rate Risk

The following video from Ryan O'Connell, CFA, FRM provides an excellent visual and numerical comparison of the actual price versus the duration approximation.

First, watch the section on calculating bond prices for different yields to understand how the data for the graph is generated. Next, watch the main analysis from graphing the relationship. Pay close attention to the visual comparison between the blue line (Actual Price) and the orange line (Duration-Implied Price). The presenter clearly explains how the linear relationship of duration fails to capture the true curvature.

The video clearly illustrates the core problem: duration assumes a linear world, but the bond market is curved. For a risk manager or trader, this error is not just a theoretical curiosity. Relying solely on a duration-based hedge in a volatile market would lead to predictable losses, as the hedge would consistently underperform against actual price movements.

The larger the yield change, and the more curved the price-yield relationship (a concept we will soon formalize as convexity), the more inadequate the linear approximation becomes.

Conclusion

This lesson has demonstrated the fundamental limitation of using Modified Duration alone to estimate bond price changes. We've established, both visually and mathematically, that this first-order, linear approximation is only accurate for small changes in yield.

Key Takeaways:

The true price-yield relationship for a standard bond is inverse and convex (non-linear).
Modified Duration provides a linear approximation of this relationship, equivalent to the first-order term in a Taylor series expansion.
For any significant change in yield (either up or down), the duration approximation underestimates the bond's actual price due to the curve's convexity. This results in underestimating price gains when yields fall and overestimating price losses when yields rise.

The error we've identified is not random; it's a direct result of ignoring the curvature of the price-yield function. In our next lesson, we will address this by formally defining and deriving Convexity, the second derivative of the price-yield function. By adding this second-order term to our approximation, we can account for the curve and dramatically improve the accuracy of our interest rate risk models.

Recommended resources

Introduction

The Non-Linear Reality of the Price-Yield Curve

Taylor Series: The Mathematical Framework

Visualizing and Quantifying the Error

Conclusion