Why Modified Duration Has a Negative Price-Yield Sign
Summary
The note explains the negative sign in the first-order relationship between a bond’s price change and its yield change. Modified duration is conventionally positive for an ordinary bond, while bond prices and yields move in opposite directions: a yield increase corresponds to a price decrease, and vice versa. The negative sign expresses that inverse relationship.
The explanation is conceptual and does not derive duration from bond cash flows or discuss convexity, so it is limited to interpreting the sign in the stated approximation. It offers no empirical evidence or worked numerical example; the takeaway is the conventional direction of the price response for a normal bond.
Key ideas
- Bond prices and yields generally move in opposite directions.
- The negative sign represents that inverse price-yield relationship.
- Modified duration is conventionally positive for a normal bond.
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Full text
# Why the negative sign in modified duration relationship # Why the negative sign in modified duration relationship If $P$ is price, $D$ modified duration and $y$ yield then we have the relationship, $$dP=-D \cdot P \cdot dy$$ Why is there a minus sign and what does correspond to? ## Answer by Phil H (score 2, accepted) https://quant.stackexchange.com/a/20619 The price-yield relationship is negatively correlated; when prices go down, the implied yield goes up. The minus sign allows the modified duration to be positive for a normal bond.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.