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CFA® Level I · Fixed Income · free lesson
Yield-Based Bond Duration Measures and Properties
Duration is the single most used number in bond risk management: it tells a portfolio manager, a bank treasurer or a pension trustee how much value is at stake when yields move. This module turns Macaulay duration into price-sensitivity measures (modified, approximate modified and money duration, and the PVBP), shows how to apply them to bonds and positions, and explains why maturity, coupon and yield level make some bonds far riskier than others.
What you will learn
1. Calculate modified duration from Macaulay duration and interpret it as the approximate percentage change in a bond's full price for a given change in yield to maturity.
- ModDur = MacDur / (1 + r), where r is the yield per period.
- Estimated %ΔPV(full) = -AnnModDur x ΔYield, with ΔYield in decimal form.
- Annualize correctly: divide by (1 + r/m) with the periodic yield, then by m if duration is in periods.
- Modified duration is a linear estimate: good for small yield changes, biased for large ones.
- It assumes cash flows are fixed, so it is not appropriate for bonds with embedded options.
2. Estimate approximate modified duration by repricing a bond for equal upward and downward yield changes, and use the estimate to forecast price changes.
- ApproxModDur = (PV- minus PV+) / (2 x ΔYield x PV0).
- The denominator uses 2 x ΔYield because the yield range runs from -ΔYield to +ΔYield.
- Shift the yield up and down by the same small amount; the central difference cancels most curvature.
- Approximate Macaulay duration = ApproxModDur x (1 + r).
- Effective duration applies the same formula to benchmark-curve shifts priced with a model.
3. Calculate and interpret money duration and the price value of a basis point (PVBP) for an individual bond and for a bond position.
- Money duration = AnnModDur x full price (per 100 par or for the whole position).
- Estimated currency change = -money duration x ΔYield.
- PVBP = (PV- minus PV+)/2 for 1 bp shifts, or about money duration x 0.0001.
- Use market value (full price), not par, when sizing exposure.
- Hedges are commonly sized by matching PVBPs; money durations add across positions.
4. Explain how a bond's maturity, coupon rate, yield level and cash-flow features affect its duration and therefore its interest rate risk.
- Longer maturity usually raises duration; higher coupon and higher yield lower it.
- Zero-coupon bond: Macaulay duration = maturity.
- Perpetuity: Macaulay duration = (1 + r)/r; modified duration = 1/r.
- Long-maturity deep-discount bonds are the exception where duration can fall as maturity increases.
- Floating-rate notes have low rate duration (about the time to the next reset); embedded call options shorten duration.
Module at a glance
- Modified duration = Macaulay duration / (1 + periodic yield); it estimates the % price change per unit change in yield.
- Estimated %ΔPV(full) = -AnnModDur x ΔYield; the sign is negative because price and yield move inversely.
- Approximate modified duration = (PV- minus PV+) / (2 x ΔYield x PV0), computed from repricing.
- Money duration = AnnModDur x full price; PVBP is about money duration x 0.0001.
- Size hedges by matching PVBPs and always use market value, not par.
- Duration rises with maturity and falls with coupon and yield; zero-coupon duration equals maturity.
- Perpetuity Macaulay duration = (1 + r)/r; floating-rate notes have duration close to the time to the next reset.
This free module has 4 lessons, 14 practice questions with full explanations.