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CFA® Level I · Fixed Income · free lesson
Curve-Based and Empirical Fixed-Income Risk Measures
Yield-based duration works only when a bond's cash flows are fixed and every yield moves together. Callable bonds, mortgage-backed securities, curve twists and credit spreads break those assumptions. This module introduces the measures practitioners use instead: effective duration and convexity from benchmark-curve shifts, key rate durations for changes in curve shape, and empirical duration estimated from how bond prices actually behave.
What you will learn
1. Explain why effective duration and effective convexity are the appropriate risk measures for bonds whose cash flows depend on interest rates, and calculate them from model prices.
- Use effective (curve-based) measures when cash flows change with rates: callables, putables, MBS.
- EffDur = (PV- minus PV+) / (2 x ΔCurve x PV0), with prices from a model after shifting the benchmark curve.
- EffCon = (PV- + PV+ - 2 x PV0) / ((ΔCurve)^2 x PV0).
- Callable bonds and MBS can have negative effective convexity when rates are low.
- Effective measures depend on model assumptions (volatility, prepayment speeds).
2. Estimate the percentage price change of a bond for a parallel shift in the benchmark yield curve using effective duration and effective convexity, and interpret negative convexity.
- %ΔPV ≈ -EffDur x ΔCurve + 0.5 x EffCon x (ΔCurve)^2.
- Negative effective convexity reduces gains when rates fall and enlarges losses when rates rise.
- Callable bonds and MBS holders are short an option and earn extra yield for bearing negative convexity.
- A callable bond far out of the money behaves like a straight bond; its convexity turns negative as the call nears.
- Putable bonds have positive convexity; the put shortens effective duration when rates rise.
3. Define key rate durations and use them to estimate the price effect of non-parallel changes in the benchmark yield curve.
- Key rate duration = sensitivity to a shift in one maturity point of the benchmark curve.
- The sum of key rate durations equals effective duration.
- %ΔPV ≈ -Σ (KeyRateDur_k x Δr_k) for any change in curve shape.
- Equal-duration portfolios can have very different key rate profiles (bullet versus barbell).
- Key rate durations reveal shaping risk and guide maturity-specific hedging.
4. Explain how changes in benchmark yields and credit spreads combine to move the price of a credit-risky bond, and distinguish empirical duration from analytical duration.
- Credit-risky yield = benchmark yield + spread; price reacts to the sum of the two changes.
- Analytical duration assumes benchmark and spread changes are independent.
- Empirical duration comes from regressing actual price changes on benchmark yield changes.
- Benchmark yields and credit spreads are often negatively correlated in stress (flight to quality).
- High-yield bonds: empirical duration well below analytical; government bonds: about equal.
Module at a glance
- Use effective duration and convexity when cash flows depend on rates: callable, putable and mortgage-backed securities.
- Effective measures shift the benchmark curve and reprice with a model; the formulas mirror the approximate yield-based ones.
- Negative effective convexity (callables near the call price, agency MBS) shrinks gains and enlarges losses.
- Key rate durations measure exposure to individual maturity points and sum to effective duration.
- For non-parallel shifts: %ΔPV ≈ -Σ KRD_k x Δr_k.
- A credit-risky bond's price responds to benchmark change plus spread change.
- Empirical duration, estimated from data, is below analytical duration for high-yield bonds because spreads widen when benchmark yields fall.
This free module has 4 lessons, 13 practice questions with full explanations.