Which of these is a reasonable approximation of bond sensitivity using duration?

Question

BlackTom AI · Accepted Answer

We are asked to identify a reasonable approximation of bond sensitivity using duration.

First, recall what duration represents: approximately how much the price of a bond moves for a small change in yield. In standard (percentage) terms, the Macaulay or modified duration relates ΔP/P to Δy via ΔP/P ≈ -D Δy, where D is duration and y is the yield (in decimal form).

To translate this into an expression for the price sensitivity dP/dy or a scaled sensitivity, we can rearrange the relation. Since ΔP ≈ -D P Δy, the sensitivity with respect to yield is dP/dy ≈ -D P. If we denote the current price by B (i.e., P ≈ B), then dP/dy ≈ -D B.

Sometimes, practitioners express a per-unit-change-in-yield sensitivity of price relative to price, i.e., (dP/dy) / P ≈ -D. To convert this to a form that uses the price itself in the denominator, one common representation is -(D) / (1+y) times something that accounts for the yield compounding, yielding a form like -(D/B) / (1+y) as a rough approximation in certain discrete compounding conventions.

Now, evaluate each option:

- Option: -D/B * (1+y)
Explanation: This multiplies by (1+y) rather than dividing by it and therefore magnifies the sensitivity rather than applying the standard inverse relationship implied by duration. It also places yield in the numerator, which is contrary to the usual negative relationship between price and yield. This is not consistent with duration-based approximations.

- Option: -D*B/(1+y)
Explanation: This expression has the price B in the numerator, which would imply sensitivity grows with the price, opposite to the typical dP/dy ≈ -D P form where sensitivity scales with price but included as a simple product with D. Additionally, dividing by (1+y) changes the scaling in a way that is not aligned with the standard relative change formula ΔP/P ≈ -D Δy. Overall, this does not align with the conventional duration-based sensitivity.

- Option: -(D/B) / (1+y)
Explanation: This form places duration D in the numerator and price B in the denominator, then divides by (1+y). It captures a ratio-style sensitivity that resembles a scaled (per-price) change, and the division by (1+y) reflects a discretized adjustment for yield. This matches the idea of a sensitivity measure expressed relative to price with a yield- adjustment factor and is the one that aligns with common compact representations used in certain yield-conversion conventions.

- Option: -(D*1000 / B) / (1+y)
Explanation: This is essentially the same structure as the previous option but with an extraneous factor of 1000 in the numerator. The inclusion of 1000 is arbitrary and numerically changes the sensitivity in a way that is not justified by standard duration-based approximations, making it incorrect.

In summary, the logic of duration-based sensitivity leads to a form that scales with duration and price and includes an adjustment for yield, without introducing unnecessary constants or inverted yield factors. Among the given choices, -(D/B) / (1+y) is the one that best aligns with these principles, while the other options distort the traditional relationships between price, yield, and duration.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 13. Quiz for Class 13

Which of these is a reasonable approximation of bond sensitivity using duration?

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