How much should you trust the idea that 'a 1% interest rate change moves price by the duration'?—Tracking the quadratic term of convexity with numbers and distinguishing …

Introduction

When explaining the interest rate risk of bonds, a first-order approximation is widely used: ‘If the modified duration is 8, a 1% rise in interest rates will cause the price to fall by approximately 8%.’ According to the Ministry of Finance’s ‘JGB Interest Rate Information,’ as of September 17, 2026, the yield on 10-year JGBs is 2.993%, and 4.047% for 30-year bonds. At its Monetary Policy Meeting on September 18, the Bank of Japan decided to raise the target for the uncollateralized overnight call rate from around 1.0% to around 1.25% (the new policy applies from September 24).

This first-order approximation is sufficiently accurate for small fluctuations. However, it remains an approximation, and the error grows in proportion to the square of the interest rate change. If the magnitude of change is doubled, the error becomes approximately four times larger; if tripled, nine times larger. The indicator that measures this quadratic term is convexity, which corresponds to the quantity captured by the second derivative of the relationship between price and yield.

Approximation error grows with the square of the fluctuation magnitude

The error of a first-order approximation has two properties. First, the error only occurs in one direction. For standard coupon-bearing bonds, the price curve is convex, so calculations based solely on duration underestimate price increases and overestimate price decreases. Second, the magnitude of the error changes significantly depending on the remaining maturity. The quadratic term that was negligible for a 10-year bond becomes a size that cannot be ignored for a 30-year bond. The phrase ‘how many percent for 1%’ is not an explanation that holds with the same precision for all maturities.

Furthermore, even for two combinations with the same duration, the quadratic quantity differs significantly. A combination of short-term and ultra-long-term bonds can have a convexity nearly twice as large as a single medium-term bond, even if their modified durations are identical. If you discuss price movements by aligning them with a single number called duration, this difference disappears from view.

For products where the sign is reversed, the direction of asymmetry is inverted

The discussion so far assumes bonds with positive convexity. For bonds where the issuer has an option for early redemption, or for securities backed by mortgages, the lower the interest rate, the faster the redemption, placing a ceiling on price increases. It is near this ceiling that the convexity, as seen by the holder, turns negative. This is a reverse asymmetry where the benefits of falling interest rates are curtailed, while the impact of rising interest rates is felt in full. Positively speaking, the yield is usually increased by that amount, and the price of the ceiling is already baked into the price. Negatively speaking, the explanation that ‘if interest rates fall, the price will rise’ changes to an explanation with an upper limit.

This structure is also carried over directly into the bond portions of insurance products. This applies to the ultra-long-term bonds held in general accounts, the surrender values of products with market value adjustments, and the bond-type special accounts of variable insurance. While they are all the same in that ‘value moves when interest rates move,’ the quadratic nature of how they move is different.

My personal conclusion is not to discard the first-order approximation, but to distinguish ‘in which situation and in which direction’ that approximation fails. There are three axes for distinction: the magnitude of fluctuation, the presence or absence of an option, and the shape of the curve. If you hear the duration, simply asking for the sign of the convexity next will change the precision of your explanation by a level.

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Table 1: Three axes where first-order approximation fails and supplementary indicators

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