Level I · Fixed Income

Learning Module 11
Yield-Based Bond Duration Measures and Properties

Key Outcomes Summary & Practice Problems

Learning Outcomes

What you must be able to do

Curriculum Year: 2026

LOS 1

Define, calculate, and interpret modified duration, money duration, and the price value of a basis point (PVBP) — key measures of a bond's price sensitivity to changes in its own yield‑to‑maturity.

LOS 2

Explain how a bond's maturity, coupon, and yield level affect its interest rate risk — the properties of duration and their implications for bond price sensitivity.

1 · Modified Duration

Modified duration (ModDur): The slope of the price‑yield curve — measures the percentage change in a bond's full price for a given change in yield‑to‑maturity.
Formula: ModDur = MacDur / (1 + r) where r is the yield per period.
Annualized modified duration: Divide by the number of periods per year (e.g., for semiannual, divide by 2).
Interpretation: A bond with ModDur = 5 will decrease by approximately 5% for a 100 bp increase in yield, and increase by approximately 5% for a 100 bp decrease.
Approximate modified duration: Estimated by changing yield up and down by a small amount (ΔYield) and calculating the resulting prices:
ApproxModDur ≈ (PV− − PV+) / [2 × ΔYield × PV0]
Percentage price change estimate: %ΔPVFull ≈ −AnnModDur × ΔAnnYield

MODIFIED DURATION

ModDur = MacDur / (1 + r)
%ΔPV ≈ −AnnModDur × ΔYield
ApproxModDur = (PV₋ − PV₊) / [2 × ΔYield × PV₀]

2 · Money Duration and Price Value of a Basis Point (PVBP)

Money duration (MoneyDur): The product of annualized modified duration and the full price of the bond (in currency units).
MoneyDur = AnnModDur × PVFull
Currency change estimate: ΔPVFull ≈ −MoneyDur × ΔYield
Price Value of a Basis Point (PVBP): The change in the full price of a bond for a 1 bp (0.01%) change in yield.
PVBP = (PV− − PV+) / 2 (using a 1 bp shift)
Also called PV01, DV01 (dollar value of 01), or basis point value (BPV).
Relation: PVBP ≈ MoneyDur × 0.0001 (approximately)

MONEY DURATION & PVBP

MoneyDur = AnnModDur × PV^Full
PVBP = (PV₋ − PV₊) / 2
PVBP ≈ MoneyDur × 0.0001

3 · Yield Duration of Special Bonds

Zero‑Coupon Bonds

Macaulay duration = time‑to‑maturity (single cash flow). Modified duration = MacDur / (1 + r).

Perpetual Bonds

Macaulay duration = (1 + r) / r. Finite duration despite no maturity.

Floating‑Rate Notes (FRNs)

Macaulay duration = fraction of the period remaining until the next reset date: (T − t) / T. Very low duration.

4 · Properties of Duration

Maturity effect: Longer time‑to‑maturity → higher duration (generally). Exception: low‑coupon, long‑term discount bonds.
Coupon effect: Lower coupon rate → higher duration (more weight on final principal payment).
Yield effect: Lower yield‑to‑maturity → higher duration (future cash flows weighted more heavily).
Time decay: Duration decreases smoothly as time passes between coupon payments (saw‑tooth pattern — jumps up slightly after each coupon).

📈 Maturity ↑

Duration ↑ (generally)
(except low‑coupon, long‑term discount bonds)

📉 Coupon ↓

Duration ↑
(more weight on final cash flow)

📊 Yield ↓

Duration ↑
(higher PV weights on distant cash flows)

⏱️ Time passes

Duration ↓ smoothly,
jumps up slightly on coupon dates

Feature	Effect on Duration	Effect on Interest Rate Risk
Longer maturity	↑ Higher (generally)	↑ Greater price sensitivity
Lower coupon	↑ Higher	↑ Greater price sensitivity
Lower YTM	↑ Higher	↑ Greater price sensitivity
Time passes (constant YTM)	↓ Lower	↓ Less price sensitivity

5 · Quick Reference — Duration Measures

Measure	Formula	Use
Macaulay Duration	`Σ t × [PV(CFₜ)/Price]`	Weighted average time to receipt; balancing point for interest rate risk
Modified Duration	`MacDur / (1 + r)`	Estimate % price change for a given yield change
Money Duration	`AnnModDur × Price`	Estimate currency price change for a given yield change
PVBP (DV01)	`(PV− − PV+) / 2`	Change in price for a 1 bp yield change