Curve-Based and Empirical Fixed-Income Risk Measures

A callable bond's cash flows shift when rates move. That single fact rules out modified duration and forces curve-based measures.

Modified duration and convexity assume cash flows are fixed. For an option-free bond, that holds. For a callable bond, falling yields trigger refinancing risk: the issuer calls the bond and the holder loses upside. For a putable bond, rising yields let the holder put it back. For mortgage-backed securities (MBS), prepayment speeds shift with rates. The cash flows now depend on where rates go.

A bond with an embedded option also has no single yield that stays meaningful as rates move, because the expected cash flows behind that yield keep changing. So you cannot measure its risk by shifting its own yield. You shift the benchmark curve instead and let a model decide, at each new rate level, whether the option gets exercised. Effective duration and effective convexity are the most appropriate measures for these bonds because they are built exactly that way.

Common mistakes

• Using modified duration on a callable bond. Modified duration assumes fixed cash flows. The call option voids that. Always use effective duration when optionality exists. Trap: applying modified duration of 7.2 to a callable trading near its call price.
• Forgetting the ½ on the convexity term. The formula is $-\text{EffDur} \times \Delta y + \tfrac{1}{2} \times \text{EffCon} \times (\Delta y)^{2}$. Trap: writing %ΔP = −D × Δy + C × (Δy)² doubles the convexity adjustment, which turns Example 1's +2.23% into +1.83%.
• Treating convexity as always positive. Callables and MBS exhibit negative effective convexity at low yields. Trap: adding a positive convexity correction to a callable when the model output is negative. In Example 1 that gives +3.05% instead of +2.23%.

Bottom line

• Use effective duration and effective convexity whenever cash flows change with yields (callables, putables, MBS, floaters with caps/floors).
• Percentage price change: %ΔP ≈ −EffDur × Δy + ½ × EffCon × (Δy)², with Δy in decimal.
• Key rate (partial) durations measure sensitivity to a single point on the curve and sum approximately to effective duration for a parallel shift. For a curve reshaping, %ΔP ≈ −Σ KeyDur × Δr at each point.
• Analytical duration is model-derived from assumed cash flows; empirical duration is regression-derived from observed prices and yields.

Exam shortcut

When you see "embedded option" in a duration question, eliminate any answer using modified duration. When you see "curve twist", "steepener", or "butterfly", reach for key rate durations, not effective duration, and multiply each one by its own rate change before adding. If you are handed three model prices, compute EffDur and EffCon from the same three numbers, then check the convexity sign before you add the second term.

The full lesson (about 2,478 words, 17 min read) adds 4 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• curve-based and empirical risk measures

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