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Financial Instruments, Markets and Conventions / Fixed Income: Bonds, Yields, and Interest Rate Risk

Duration and Convexity: Measuring and Managing Interest Rate Sensitivity

Every portfolio manager who holds bonds is exposed to interest rate risk: when yields rise, bond prices fall. But by how much? And for which bonds does a 100 basis-point move hurt the most? The answers come from two complementary risk measures — duration and convexity — that together characterise the shape of the price-yield relationship with enough precision to manage large fixed-income portfolios.
Duration does the heavy lifting: it quantifies the first-order sensitivity of a bond's price to its yield. A bond with a modified duration of 10 loses approximately 10% of its value for every 100bp rise in yield. This linear approximation is good for small yield moves. For large moves — say, a 200bp shock — duration's linear approximation becomes inaccurate because the true price-yield relationship is curved (convex). Convexity captures that curvature and allows much better approximations of the actual price change.

Together, these two measures drive every bond trading strategy: duration-matching for immunisation, duration-neutral pairs trades, DV01 hedges, and the convexity position that benefits from large yield moves in either direction.

Learning objectives. By the end of this note you should be able to:
  • state the four price-volatility properties of an option-free bond;
  • compute the price value of a basis point (PVBP / DV01) for a bond;
  • compute Macaulay duration and modified duration from first principles;
  • use modified duration to approximate the percentage price change for a yield shift;
  • compute dollar duration and estimate the dollar price change for a yield shift;
  • explain the limitations of duration for large yield moves and how convexity corrects them;
  • use modified duration and the convexity measure together to estimate the price change for large yield moves;
  • compute portfolio duration as a market-value-weighted average of individual durations.

In this lesson

  1. Price-volatility characteristics of option-free bonds
  2. Price value of a basis point (PVBP / DV01)
  3. Macaulay duration
  4. Modified duration
  5. Dollar duration and DV01
  6. Properties of duration
  7. Convexity: the second-order correction
  8. Practical approximation of duration and convexity
  9. Portfolio duration and immunisation

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