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Estimate a bond's price change when interest rates move, using its duration and convexity.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.g

The idea

A second-order expansion estimates a bond's percentage price change for a yield change dy. Percentage price change = −D × dy + ½ × C × (dy)². Here D is modified duration and C is convexity. The duration term is negative for a yield rise and positive for a fall. The convexity term is positive whichever way yields move, because dy is squared, provided the bond has positive convexity. In money terms, multiply by the price or position value. If a question gives Macaulay duration, first convert it: modified duration = Macaulay duration ÷ (1 + y/m). Yield changes must be in decimals, so 25 basis points is 0.0025. Duration alone is accurate for small moves, and convexity matters for large ones. The frequent mistakes are subtracting the convexity term, omitting the one half, and using Macaulay duration or percentage points in place of decimals.

Formulas

Percentage price change from duration and convexity

percentage price change = −D × Δy + ½ × C × (Δy)²

D
modified duration
C
convexity
Δy
the yield change, in decimals

The convexity term is added for positive convexity whichever way yields move. Convert Macaulay to modified duration first, and keep the one half.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which expression gives the approximate percentage change in a bond's price for a small yield change dy, using modified duration D and convexity C?

  1. -D x dy - 0.5 x C x (dy)²Why not: Subtracts the convexity term; the convexity adjustment is positive for a bond with positive convexity.
  2. -D x dy + 0.5 x C x (dy)² - correct
  3. -D x dy + C x (dy)²Why not: Omits the factor one half from the second-order Taylor term.
  4. D x dy + 0.5 x C x dyWhy not: Drops the minus sign on the duration term and does not square the yield change in the convexity term.

A second-order Taylor expansion of price in yield gives dP/P = -D x dy + 0.5 x C x (dy)². The duration term is linear and negative for a yield rise; the convexity term is always positive for positive C because dy is squared.

A harder one

A bond priced at 100 has modified duration 8.0 and convexity 75. The yield falls by 100 basis points. Using duration and convexity, the estimated new price is closest to:

  1. 108.38 - correct
  2. 107.62Why not: Subtracts the convexity term; for a yield fall the convexity adjustment adds to the price gain.
  3. 108.75Why not: Omits the factor of one half in the convexity term.
  4. 108.00Why not: Uses duration only and ignores the convexity adjustment.

Percentage change = 8.0 x 0.01 + 0.5 x 75 x 0.01² = 8.00% + 0.375% = 8.375%. New price = 100 x (1 + 0.08375) = 108.38.

The hardest tier

A position has market value USD 25 million, modified duration 6.2 and convexity 58. Yields fall by 150 basis points. Using duration and convexity, the estimated change in the position's value is closest to:

  1. a gain of USD 2,325,000Why not: Uses duration only and omits the convexity gain.
  2. a gain of USD 2,161,875Why not: Subtracts the convexity term; it adds to the gain whichever way yields move.
  3. a gain of USD 2,488,125 - correct
  4. a gain of USD 2,651,250Why not: Omits the factor one half in the convexity term.

Return = 6.2 x 0.015 + 0.5 x 58 x 0.015² = 9.30% + 0.6525% = 9.9525%. Change = 25 million x 0.099525 = 2,488,125.

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