Effective duration of putable bond.

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pivpomars

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Hello,
I have a question concerning the putable bond. The book 4, Schewer 2015, Los 44j said that:
“effective duration of putable < effective duration of straight bond”

I think it’s fault because:
Value( Putable) = Value (Straight) + Value (Put)
Because effective duration of X = Variation of X / Variation of Interest Rate ( effectiveDuration = dX/dRate) , so effectiveDuration ( Putable) = effectiveDuration (Straight) + effectiveDuration (Put)
And we know that: If Interest Rate increases, Bond’s value decreases, so, Put increases
=> effectiveDuration (Put) > 0
Thus,
effectiveDuration ( Putable) = effectiveDuration (Straight) + effectiveDuration (Put) > effectiveDuration (Straight)
Am I wrong?
 
I just think that
EffectiveDuration(X) = 1/X * (dX/dI) (I - Interest Rate)
So
EffectiveDuration(Putable) = 1/V(Putable) * d(Putable)/dI = 1/V(Putable) *[d(Straight)/dI + d(Put)/dI ]
= V(Straight)/V(Putable) * EffectiveDuration(Straight) + 1/V(Putable) *d(Put)/dI
I put: Delta = EffectiveDuration(Putable) - EffectiveDuration(Straight)
So, Delta = [ V(Straight)/V(Putable) - 1] * EffectiveDuration(Straight) + 1/V(Putable) *d(Put)/dI
So, there are 2 effects on the delta
- The first one is [ V(Straight)/V(Putable) - 1] * EffectiveDuration(Straight) makes Delta becomes negative
- The second one is 1/V(Putable) *d(Put)/dI makes Delta becomes positive
Maybe Schweser (and CFA Curiculum) suppose that the first effect is stronger than the second, so, effectiveDuration ( Putable) < effectiveDuration ( Straight)
 
When interest rates are low, the price of the putable bond is roughly equal to that of the straight bond, but when interest rates are high, the price of the putable bond is higher than that of the straight bond (because the putable bond has a floor value equal to the put price, while the straight bond has a floor value of zero). Thus, as interest rates rise, the price of the putable bond drops less than that of the straight bond; i.e., its effective duration is less.
 
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