Curve-Based and Empirical Fixed-Income Risk Measures

Interest rate risk measured against a benchmark yield curve instead of a bond's own yield: effective duration and effective convexity for bonds with embedded options, the negative convexity of callable bonds, price estimates for a curve shift, key rate durations for non-parallel moves, and why empirical duration can beat analytical duration for bonds with credit risk.

1. 1. Effective duration: why curve duration replaces yield durationWhen a bond's cash flows depend on future interest rates, its yield-to-maturity is not well defined, so we measure its price sensitivity to a parallel shift in a benchmark yield curve instead: effective duration.Free13 min
2. 2. Effective convexity: callable versus putable bondsAn embedded call caps a bond's price gains when rates fall and can make its effective convexity negative, while an embedded put cushions price losses when rates rise and keeps effective convexity positive.Video · 7 min13 min
3. 3. Estimating price changes with effective duration and convexityPlug effective duration and effective convexity into the familiar duration-plus-convexity formula, with ΔCurve in place of ΔYield, to estimate the percentage change in a bond's full price for a shift in the benchmark curve.12 min
4. 4. Key rate duration and yield curve shape riskKey rate durations split effective duration across points of the benchmark curve, so you can estimate how a bond or portfolio reacts when the curve steepens, flattens or twists rather than shifting in parallel.Video · 7 min14 min
5. 5. Empirical versus analytical durationAnalytical duration comes from pricing formulas that treat benchmark yields and spreads as independent; empirical duration is estimated from historical price data and captures the fact that spreads often widen just as government yields fall.11 min