The duration rule always ________ the value of a bond following a change in its yield

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Question restatement: The duration rule always ________ the value of a bond following a change in its yield

Option A: Underestimate.
Option B: Provides an unbiased estimate.
Option C: Overestimates.
Option D: The estimated price may be biased either upward or downward, depending on whether the bond is trading at a discount or a premium.

Analysis of each option:
- Option A (Correct): Underestimate.
  The standard duration approximation for a small change in yield uses the formula price change ≈ -Duration × Δy. This is a linear (first-order) approximation. Because bonds exhibit convexity, the actual price change includes a positive 0.5 × Convexity × (Δy)² term. When yields move, the convexity term causes the real price change to differ from the linear estimate. For yield decreases (Δy negative), the convexity term adds positively, making the actual price rise larger than the duration estimate, which means the duration rule tends to underestimate the price change in magnitude for downward yield moves. Similarly, for yield increases (Δy positive), the convexity term partially offsets the price decline, but the key point is that the linear rule is not perfect; the statement as given is treated in many exam contexts as stating that the duration rule underestimates the magnitude of the price change in typical benchmark discussions. Therefore, “Underestimate” is the best-supported completion among the choices.
- Option B (Incorrect): Provides an unbiased estimate.
  In reality, the duration rule is a linear approximation and inherently ignores convexity. The presence of convexity means the approximation is not unbiased in general; the actual price change deviates from the linear estimate, with the direction and size of the bias depending on the yield movement and the bond’s convexity. So it is not guaranteed to be unbiased.
- Option C (Incorrect): Overestimates.
  This would assert that the linear duration rule errs on the side of predicting too large a price change in magnitude in all cases. However, because convexity adds to the price change when yields fall and subtracts from the decline when yields rise, the bias is not simply a universal overestimation. The common teaching is that the linear estimate is typically too small in magnitude for downward yield moves (i.e., underestimates), not consistently overestimates across all scenarios.
- Option D (Incorrect): The estimated price may be biased either upward or downward, depending on whether the bond is trading at a discount or a premium.
  The bias direction is driven by yield movement (Δy) and the bond’s convexity, not by whether the bond currently trades at a discount or premium. The idea that the price bias would depend on the price level (discount/premium) misattributes the source of the error; convexity interacts with the sign of Δy, not the origination of the price relative to par. Therefore this option mischaracterizes the source of the bias.

Summary of reasoning:
The duration-based price change is a first-order (linear) approximation that ignores convexity. Convexity causes the actual price change to diverge from the duration estimate, with the direction and magnitude of the bias depending on the yield movement, not strictly on whether the bond is at a discount or premium. Among the given choices, only Underestimate correctly captures the typical bias introduced by ignoring convexity in the duration rule for bond price changes after yield changes.

BU.232.750.81.FA25 Final Exam Fall 2025- Requires Respondus LockDown Browser

The duration rule always ________ the value of a bond following a change in its yield

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