CFA L1 / FI / M13. Curve Risk
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

About: For bonds with embedded options. Numerically simulates parallel curve shifts and measures price response. Required for callable/putable/MBS where CFs aren't fixed.Tóm tắt: Cho TP có embedded option. Mô phỏng parallel shift và đo phản ứng giá. Cần cho callable/putable/MBS.
\[ \text{EffD} = \frac{P_{-} - P_{+}}{2 \cdot P_0 \cdot \Delta\text{Curve}} \]

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

About: Same numerical-shift method as effective duration but second-order. Captures price curvature for option-embedded bonds.Tóm tắt: Cùng phương pháp effective duration nhưng bậc 2. Đo độ cong giá cho TP có option.
\[ \text{EffC} = \frac{P_{-} + P_{+} - 2P_0}{P_0 \cdot (\Delta\text{Curve})^{2}} \]
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

About: Sensitivity to a single key maturity (2y, 5y, 10y, 30y) holding others constant. Sum ≈ effective duration. Captures NON-PARALLEL curve shifts.Tóm tắt: Nhạy cảm với 1 kỳ hạn cụ thể. Tổng ≈ effective duration. Bắt được non-parallel shift.

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

About: Statistical estimate from regressing observed price changes on yield changes. Captures real market relationship including liquidity and credit factors.Tóm tắt: Ước lượng thống kê từ hồi quy thay đổi giá vs thay đổi yield. Phản ánh quan hệ thị trường thực.

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