Module 13 · Fixed Income

Curve-Based and Empirical Fixed-Income Risk Measures

EN: Effective duration/convexity, key-rate duration, and empirical duration.
VN: Effective duration/convexity, key-rate duration, empirical duration.

1. Effective Duration Core

\[ \text{EffD} = \frac{P_{-} - P_{+}}{2 \cdot P_0 \cdot \Delta\text{Curve}} \]

Components

P₋ Price after parallel curve shift down.
P₊ Price after parallel curve shift up.
ΔCurve Magnitude of shift (in decimal).

When to use: bonds with embedded options or contingent CFs (callable, putable, MBS) — yield-based duration assumes fixed CFs which aren't valid here.

2. Effective Convexity Core

\[ \text{EffC} = \frac{P_{-} + P_{+} - 2P_0}{P_0 \cdot (\Delta\text{Curve})^{2}} \]

Practice problem

P0 = 100, P+ = 99, P− = 101.10, Δy = 50bp. Compute EffC.

Show solution

= (101.10 + 99 − 200) / [100 × (0.005)²]

= 0.10 / 0.0025

EffC = 40

3. Key-Rate Duration Core

Sensitivity of price to a change in a single key maturity (e.g. 2y, 5y, 10y, 30y) holding all others constant. Sum of all key-rate durations ≈ effective duration. Useful for analyzing non-parallel curve shifts (steepening, flattening, twist).

4. Empirical Duration Concept

Statistical estimate from regressing observed price changes on yield changes — captures the actual market relationship including liquidity and credit factors.

Practice problem Practice

Practice problem

A callable bond's price is $98.50. After +25bp parallel curve shift it falls to $97.10; after −25bp shift it rises to $99.80. Compute effective duration.

Show solution

EffD = (P− − P+) / (2 × P0 × Δcurve)

= (99.80 − 97.10) / (2 × 98.50 × 0.0025)

= 2.70 / 0.4925

Effective duration ≈ 5.48