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 assumes 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, prepayment speeds shift with rates. Cash flows are no longer fixed, they are state-dependent. That state-dependence is why effective duration and effective convexity, not their modified cousins, are the most appropriate interest rate risk measures once a bond carries an embedded option.

KEY: Effective duration uses a pricing model (typically a binomial interest rate tree or Monte Carlo for MBS) to reprice the bond after shocking the benchmark curve up and down. The model lets cash flows respond to the new rate environment. Modified duration cannot do this.

The effective duration formula:

$$\text{EffDur}=\frac{P_{-}-P_{+}}{2\times P_0\times \Delta y}$$

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 $-\text{EffDur}\times \Delta y+\frac{1}{2}\times \text{EffCon}\times (\Delta y{)}^2$. Trap: writing %ΔP = −D × Δy + C × (Δy)² and overstating the convexity adjustment by 2×.
• 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.

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.
• 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. When the bond is high-yield and the question contrasts "model" vs. "observed" sensitivity, expect empirical duration to be lower than analytical, that is the structural pattern from spread-yield correlation.

The full lesson (about 1,683 words, 11 min read) adds 2 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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