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Lesson 1 of 5 · 13 min
Effective duration: why curve duration replaces yield duration
When a bond's cash flows depend on future interest rates, its yield-to-maturity is not well defined, so we measure its price sensitivity to a parallel shift in a benchmark yield curve instead: effective duration.
In short
- Yield duration (Macaulay, modified, approximate modified) assumes the cash flows are certain and only the bond's own YTM moves.
- A callable or putable bond, or a mortgage-backed security, has cash flows that depend on whether an option is exercised, so it has no single well-defined YTM.
- Effective duration = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀). It is a curve duration: the shift is in the benchmark curve, not in the bond's yield.
- PV₊ and PV₋ come from an option pricing model in which only the benchmark curve is moved; call schedule, credit spread and volatility are held constant.
- Effective duration also works for option-free bonds, but it usually differs slightly from modified duration; the two are equal only if the yield curve is flat.
Where yield duration breaks down
Every yield duration statistic from the earlier modules is built on one assumption: the bond will pay a known schedule of cash flows on known dates. Change the YTM, rediscount the same cash flows, and you have the price sensitivity.
A bond with an embedded option breaks that assumption. A callable bond may be repaid early if rates fall, because the issuer can refinance more cheaply. A putable bond may be sold back to the issuer if rates rise. Which cash flows arrive, and when, depends on where rates go.
So there is no single, well-defined yield-to-maturity. Picking one scenario, such as yield-to-worst, and computing its duration describes only one possible outcome out of many. Macaulay and modified duration are therefore not appropriate risk measures for these bonds.
Think of a callable bond as a bond whose maturity date moves around with interest rates. A duration that assumes a fixed maturity cannot capture that.
Effective duration: shift the curve, not the yield
The fix is to ask a different question: how much does the bond's price change when the benchmark yield curve (for example, the government par curve) shifts in parallel? That sensitivity is effective duration. Because it is defined against a curve, it is a curve duration statistic rather than a yield duration statistic.
The arithmetic mirrors approximate modified duration. Shift the benchmark curve down by ΔCurve and reprice to get PV₋; shift it up by the same amount to get PV₊; PV₀ is today's full price. Then .
The difference is where the prices come from. For a bond with an option, PV₊ and PV₋ cannot be found by simple discounting; they come from an option pricing model that values the bond and its option together.
What goes into the pricing model
The model needs five inputs. The analyst freezes the first four and moves only the fifth, up and down, to generate PV₊ and PV₋.
- The length of any call protection period.
- The schedule of call (or put) dates and prices.
- An assumed credit spread over the benchmark (including any liquidity spread).
- An assumed interest rate volatility.
- The level of benchmark rates, e.g. the government par curve. This is the input that is shifted.
A lower effective duration can be read as a shorter expected life: the weighted average time until cash flows arrive shrinks when early repayment becomes more likely.
Effective versus modified duration for option-free bonds
Effective duration is essential for bonds with options, but it can also be computed for a plain option-free bond as a supplement to yield duration. Expect a small difference from modified duration.
The reason: when the model shifts the government par curve in parallel, the spot (zero) curve also moves, but not by the same amount at every maturity. So the bond's price change is not quite the same as if its own YTM had moved by ΔCurve.
The gap narrows when the yield curve is flatter, the bond's maturity is shorter, and the bond is priced closer to par. It disappears only in the rare case of a perfectly flat yield curve.
| Approximate modified duration | Effective duration | |
|---|---|---|
| What is shifted | The bond's own YTM (ΔYield) | The benchmark yield curve (ΔCurve) |
| Type | Yield duration | Curve duration |
| Cash flows | Assumed certain | May depend on rates (options, prepayments) |
| Source of PV₊, PV₋ | Discounting at the shifted YTM | Option pricing model, curve shifted |
| Use | Option-free bonds | Bonds with embedded options and MBS; also option-free bonds |
Formula
- Effective duration
PV₋, PV₊ = full prices after lowering and raising the benchmark curve by ΔCurve (decimal, e.g. 25 bp = 0.0025). PV₀ = current full price.
Worked examples
Example 1 Effective duration of a callable bond
A callable corporate bond has a full price of 101.25. An option pricing model gives 99.10 when the government par curve is shifted up 30 bp and 103.05 when it is shifted down 30 bp. What is the bond's effective duration?
Example 2 Yield duration and curve duration of an option-free bond
An option-free government-guaranteed bond is priced at 100.000. Moving its own YTM by ±10 bp gives prices of 99.355 and 100.648. Moving the government par curve by ±10 bp gives 99.342 and 100.661. Compare the two duration measures and explain the gap.
Example 3 Choosing the right measure
An analyst must report interest rate risk for three holdings: (1) a 10-year option-free government bond, (2) a 10-year corporate bond callable after 3 years, (3) a residential mortgage-backed security. Which measure is most appropriate for each?
On the exam
Expect a direct effective duration calculation from PV₀, PV₊, PV₋ and a bp curve shift, and concept items on why yield duration fails for callable bonds and MBS, and which input is shifted in the pricing model.
- Using modified duration or yield-to-worst duration for a callable bond. Its cash flows are uncertain, so use effective duration.
- Forgetting the 2 in the denominator, which doubles the answer.
- Entering ΔCurve as 25 instead of 0.0025.
- Assuming effective and modified duration of an option-free bond are always equal. They are equal only with a flat yield curve.
Check yourself
Q1A callable bond's full price is 99.20. A model gives 97.40 with the benchmark curve up 25 bp and 100.85 with it down 25 bp. Its effective duration is closest to:
Q2Modified duration is not an appropriate risk measure for a callable bond mainly because:
Q3To compute effective duration, the analyst raises and lowers which input in the option pricing model?
Q4For an option-free bond, effective duration and modified duration will be identical only when:
Q5In a callable bond's effective duration calculation, an audit shows PV₊ should be higher, with PV₀ and PV₋ unchanged. The corrected effective duration will be:
Answer the 5 questions: +10 XP for each one you get right first time.
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