Module 11 · Fixed Income

Yield-Based Bond Duration Measures

EN: Macaulay, modified, and money duration; price impact of yield changes.
VN: Macaulay, modified và money duration; ảnh hưởng giá khi yield thay đổi.

1. Macaulay Duration Core

\[ \text{MacD} = \sum_{t=1}^{N} \frac{t \cdot PV(CF_t)}{P_0} \]

Components

PV(CF_t) Present value of cash flow at time t.
P₀ Bond price (sum of all PVs).

Macaulay duration = weighted-average time-to-receipt of cash flows.

Practice problem

3-year, 5% coupon, face $1,000, YTM 5% (price = par $1,000). Compute MacD.

Show solution

Weights (PV·t / P): t=1 → 47.62/1000=0.0476; t=2 → 90.70/1000=0.0907; t=3 → 906.69/1000=0.9067

MacD = 1(0.0476) + 2(0.0907) + 3(0.9067)

= 0.0476 + 0.1814 + 2.7201

MacD ≈ 2.86 years

2. Modified Duration Core

\[ \text{ModD} = \frac{\text{MacD}}{1 + \tfrac{YTM}{m}} \]

Use

m Periods per year (semi-annual = 2).
Approx. % price change ≈ −ModD × ΔYTM.

Practice problem

MacD = 6.5, semi-annual YTM = 3% (annual 6%). Compute ModD.

Show solution

ModD = 6.5 / (1 + 0.06/2)

= 6.5 / 1.03

ModD ≈ 6.31

3. Money Duration & PVBP Core

\[ \text{Money duration} = \text{ModD} \times P_0 \] \[ PVBP = \frac{\text{Money duration}}{10{,}000} = \text{ModD} \times P_0 \times 0.0001 \]

PVBP (price value of a basis point) = $ price change for 1 bp yield change.

Practice problem

ModD = 7.0, price = $980. Compute money duration and PVBP.

Show solution

Money D = 7.0 × 980 = 6,860

PVBP = 6,860 × 0.0001

Money D = $6,860; PVBP = $0.686

4. Properties of Duration Core

Maturity ↑ Duration ↑.
Coupon ↑ Duration ↓ (faster CF recovery).
YTM ↑ Duration ↓ (higher discount weights later CFs less).
Zero-coupon MacD = maturity exactly.

5. Portfolio Duration Core

\[ \text{Portfolio ModD} = \sum_{i=1}^{n} w_i \cdot \text{ModD}_i \]

Weights based on market value (= price + accrued interest).

Practice problem

60% in Bond A (ModD = 4), 40% in Bond B (ModD = 9). Compute portfolio ModD.

Show solution

= 0.60(4) + 0.40(9) = 2.4 + 3.6

Portfolio ModD = 6.0

Practice problem

Bond has Modified Duration of 6.5, price $98.50. Yield rises 25 bps. Estimate the % and $ price change.

Show solution

% change ≈ −6.5 × 0.0025 = −0.01625 = −1.625%

$ change ≈ −1.625% × 98.50

≈ −$1.601 → new price ≈ $96.90