Explain effective duration and the cases that require it
The idea
Modified duration is computed analytically from a bond's fixed cash flows, which fails the moment the cash flows themselves depend on the level of rates. For a callable, putable or mortgage-backed security, effective duration must be used instead: the price is revalued under a small upward and downward shift of the benchmark curve using a model, and the sensitivity is measured from those two prices. Because the option changes the cash flows, effective duration can be far below modified duration, and for a mortgage security it can even turn negative when prepayments accelerate.
Formulas
Effective duration
EffDur = (PVdown − PVup) ÷ (2 × Δcurve × PV₀)
- PVdown
- the price after the benchmark curve is shifted down
- PVup
- the price after it is shifted up by the same amount
- Δcurve
- the size of that shift, as a decimal
- PV₀
- the price before either shift
For bonds whose cash flows change with yields — callable, putable, mortgage-backed — where modified duration does not apply because there is no one yield to maturity to differentiate. It shifts the whole curve and reprices instead.
A practice question
For which bond must effective duration be used rather than modified duration?
- A fixed-rate government bondWhy not: Its cash flows are fixed and known, which is exactly the condition under which the analytic formula works.
- A callable corporate bond — correct
- A zero-coupon bondWhy not: The simplest case of all — one cash flow at one date, with Macaulay duration equal to maturity.
The call makes the cash flows depend on the level of rates, so an analytical formula built on fixed cash flows no longer applies.
A harder one
A bond is revalued at 101.2 after a 25 basis point fall and 98.9 after a 25 basis point rise, from a price of 100. Roughly what is its effective duration?
- 2.3Why not: The price range itself, 101.2 − 98.9. It still has to be divided by twice the yield shift and by the starting price.
- 4.6 — correct
- 9.2Why not: The range divided by one shift rather than two. The denominator is 2 x 0.0025 because the two revaluations straddle the starting point.
The price range, 101.2 − 98.9 = 2.3, over twice the shift and the starting price, 2 × 0.0025 × 100 = 0.5, gives 2.3 ÷ 0.5 = 4.6.
The hardest tier
Why can a mortgage-backed security's effective duration turn negative?
- Because its coupon can fall below zeroWhy not: No pass-through coupon goes negative. The negative sign belongs to the *duration*, meaning price falls as rates fall.
- Because its credit spread narrows as rates fallWhy not: A narrowing spread would raise the price, which is the opposite of the effect being described.
- Because prepayments accelerate as rates fall — correct
Returned principal must then be reinvested at the new lower rates. Enough prepayment can make the security worth less when rates fall, which is a price response with the opposite sign to an ordinary bond's.
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