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Modified duration

S

Studystuff

Member
Hello,
I was hoping you could help me with the concept of modified duration in chapter 13. My confusion is around the the use of Yo and Y.

I understand how when a bond has zero coupons or annual coupons then m will equal 1 in the modified duration formula so no worries here.

My question is if we have a bond with semi annual coupons. Yo will be calculated by determining the bond yield (as done in the example on page 9)

However for the modified duration do we divide this annual effective rate (Yo) by 2? It doesnt seem to make sense to divide an effective rate by two. Should we use maybe a nominal rate convertible semi annually in this formula (for y) and then use an annual effective rate in the main price volatiliy conversion formula (on the bottom of page 9?)

Thanks!
 
Hi,
The key is that, if a bond had semi-annual coupons, we would refer to its yield using a semi-annually compounding rate. So if such a bond has a yield of 2%pa, that means 1% per 6 month. It would not be described using an annual effective rate (which as you say, would not be divided by 2 at any point).
So if a semi-annual bond, with annual GRY of 2%, had a duration (macaulay duration) of 7, then its Mod duration would be 7/(1+(0.02/2) )
If you want to derive it, try writing the formula for the price of such a bond (P = (C/2) * ( (1+(i/2)^-0.5 + (1+(i/2)^-1 + ...) ) and then calculate (-1/P) * (dP/di). You will probably get: Duration /(1 + (0.02/2) ) where duration is the typical macaulay duration weighted mean term formula.
 
How does this then tie in with the price volatility and yield volatility relation then ? It states in the solution to the last question in c.13 that annual effective rates must be used for this, would you have to go manipulating between these two rates to used the modified duration formula and then the price/yield volatility formula ?
 
Lets hope that when / if it comes up, the examiners are kind enough to use an annual paying bond :) PS, it has never been tested in ST5 or SP5 so we have no past paper questions to go with. But my feeling is that yes, you would have to convert in these circumstances. The modified duration of a bond can be calculated easily enough whether its annual effective and annual paying bond, or whether its a semi-annual paying bond with a semi-annual compounding GRY. But when you go to apply St.Dev(price, forward) = MD * St.Dev(yield, forward)*Y, then because the volatilities are annual, the forward yield Y would have to be annual effective as it says in Ch13.
 
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