Level I · Fixed Income

Learning Module 13
Curve-Based and Empirical Fixed-Income Risk Measures

Key Outcomes Summary & Practice Problems

Learning Outcomes

What you must be able to do

Curriculum Year: 2026

LOS 1

Explain why effective duration and effective convexity are the most appropriate measures of interest rate risk for bonds with embedded options — because yield duration assumes certain cash flows, which is not true for callable, putable, or MBS bonds.

LOS 2

Calculate the percentage price change of a bond for a specified change in benchmark yield, given the bond's effective duration and convexity — using the curve‑based version of the duration + convexity formula.

LOS 3

Define key rate duration and describe its use to measure price sensitivity of fixed‑income instruments to benchmark yield curve changes — isolating sensitivity to specific maturities along the yield curve.

LOS 4

Describe the difference between empirical duration and analytical duration — analytical uses mathematical formulas; empirical uses historical data and accounts for correlation between spreads and benchmark yields.

1 · Effective Duration and Effective Convexity

Yield duration assumes cash flows are certain. For bonds with embedded options (callable, putable, MBS), cash flows are uncertain because option exercise depends on interest rates.
Effective duration (EffDur): Measures the sensitivity of a bond's price to a change in the benchmark yield curve (not its own YTM). Uses option pricing models to determine PV+ and PV−.
Effective convexity (EffCon): The second‑order curve‑based measure of interest rate risk.
Callable bonds: Have negative convexity at low yields (price appreciation is capped). Effective duration decreases as yields fall (bond is more likely to be called).
Putable bonds: Always have positive convexity. Effective duration decreases as yields rise (put option limits price depreciation).

EFFECTIVE DURATION & CONVEXITY

EffDur = (PV₋ − PV₊) / [2 × ΔCurve × PV₀]
EffCon = (PV₋ + PV₊ − 2PV₀) / [(ΔCurve)² × PV₀]

📞 Callable Bond

• Negative convexity at low yields
• Effective duration decreases as yields fall
• Price appreciation capped by call price

🛡️ Putable Bond

• Always positive convexity
• Effective duration decreases as yields rise
• Price depreciation limited by put price

2 · Bond Risk and Return Using Curve-Based Duration and Convexity

Effective duration and effective convexity are used in the same way as yield duration and convexity, but with ΔCurveinstead of ΔYield.
Percentage price change: %ΔPV ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)²
For callable bonds, negative convexity means the convexity adjustment is negative, increasing the price decline when yields rise and reducing the price increase when yields fall.
For option‑free bonds, effective duration and modified duration are similar but not identical; differences arise because a parallel shift in the par curve causes a non‑parallel shift in spot rates.
Choosing the size of the curve shift (ΔCurve) is a practical consideration; smaller shifts improve accuracy for option‑free bonds but may not be appropriate for bonds with embedded options.

CURVE-BASED PRICE CHANGE

%ΔPV ≈ −EffDur × ΔCurve + ½ × EffCon × (ΔCurve)²

3 · Key Rate Duration

Key rate duration (partial duration): Measures a bond's sensitivity to a change in the benchmark yield at a specific maturity (e.g., 2‑year, 5‑year, 10‑year rates).
Key rate duration formula: KeyRateDurk = − (1/PV) × (ΔPV / Δrk)
Sum property: The sum of all key rate durations equals the effective duration of the bond or portfolio.
Use: Identifies shaping risk — sensitivity to changes in the shape of the yield curve (steepening, flattening, twisting).
Portfolio management: A manager can tilt key rate durations to capitalize on expected changes in the shape of the yield curve.

KEY RATE DURATION

KeyRateDur_k = − (1/PV) × (ΔPV / Δr_k)
Σ KeyRateDur_k = EffDur
%ΔPV ≈ −Σ KeyRateDur_k × Δr_k

📊 Uses

• Identify yield curve shaping risk
• Over/underweight specific maturities
• Hedge non‑parallel yield curve shifts

🔑 Key Maturities

Commonly used: 0.5‑year, 2‑year, 5‑year, 10‑year, 20‑year, 30‑year rates

4 · Empirical Duration

Analytical duration: Estimated using mathematical formulas (Macaulay, modified, effective duration). Assumes benchmark yields and spreads are uncorrelated.
Empirical duration: Estimated using historical data in statistical models that incorporate factors affecting bond prices. Accounts for the correlation between benchmark yields and credit spreads.
Flight to quality: During market stress, government benchmark yields fall while credit spreads widen. This negative correlation reduces empirical duration relative to analytical duration for credit‑risky bonds.
For government bonds with little credit risk, analytical and empirical duration are similar. For corporate bonds, empirical duration is often lower than analytical duration (especially for high‑yield bonds).
Empirical duration is a more accurate measure for forecasting price changes of bonds with credit risk.

Feature	Analytical Duration	Empirical Duration
Estimation method	Mathematical formulas	Historical data / statistical models
Assumption	Benchmark yields and spreads are uncorrelated	Accounts for correlation between yields and spreads
Best for	Government bonds (low credit risk)	Corporate bonds (credit risk)
During flight to quality	Overestimates price sensitivity	Better reflects actual price behavior

5 · Quick Reference — Duration Types

Duration Type	Definition	Use Case
Macaulay Duration	Weighted average time to receipt of cash flows	Balancing reinvestment and price risk
Modified Duration	Price sensitivity to changes in own YTM	Option‑free bonds with certain cash flows
Effective Duration	Price sensitivity to changes in benchmark yield curve	Bonds with embedded options (callable, putable, MBS)
Key Rate Duration	Sensitivity to yield change at specific maturity	Measuring yield curve shaping risk
Empirical Duration	Estimated from historical data with spread correlation	Corporate bonds with credit risk