# Options reduce effective duration

Effective duration is a bond’s price sensitivity given a 100bps change in yield.

Then we are told that the durations of both callable and putable bonds cannot exceed that of a normal bond. The only intuitive way I understand this is that if a bond has embedded options it’s maturity will be either equal to or less than the normal bond’s maturity. Are we saying here that the effective duration is indeed a time measure? (like a length of time) because we had established that it is a price sensitivity, not a time measure.

In that case given embedded options how does a bond’s price sensitivity decrease?

That’s for modified duration. Is there a difference? I learnt it in L1.

Duration and its variants are all in years. Think about duration as a bond’s effective maturity. It also helps approximate price sensitivity. If the bond has an option it can either be called/put(ted) before it’s maturity which reduces its duration (effective maturity) and thereby its price sensitivity to interest rates more than an option free bond’s duration (effective maturity.) Hope this helps!

This whole duration concept has been a mystery since L1 and it is the name that confuses me. Apparently it is not strictly a time measure, and believe it or not, Wikipedia is the only source I found that actually recognizes there is a confusion in the intuition behind it.

https://en.wikipedia.org/wiki/Bond_duration

Macaulay Duration is a time/period measure. Multiplying the Macaulay Duration by 1/y will give you modified duration - which is the first derivative (in calculus terms) of the price yield function of a bond. By definition the first derivative tells you the instanteous rate of chage with respect to something else (i guess for a lack of better teminology). In the case of a bond the modified duration give us the change in price with respect to a change in yield.

Now with the effective duration and intution, as Amruth stated, is these bonds will either be called/put prior to maturity (assuming rates move). Think of a 30 year bond callable in 10 year paying a 10% coupon. If rates go to 0 this bond will most certainly now be pricing and behaving as a 10 year bond. Thus the duation will be the equivelant to a 10 year bond (which is much lower) rather than the original 30 year bond. And as we learned bonds with longer matuiries will have longer durtions (in most cases). Thus the optionality has decreased the duration of the bond.

Modified duration _ is not _ the first derivative of the price with respect to yield, as it measures _ percentage _ price change.

Dollar duration is (the negative of) the first derivative of the price with respect to yield, as it measures price change.

Modified duration is (the negative of) the first derivative of ln(price) with respect to yield, but who wants to bother with that?

My apologies for oversimplifying…

I have watched and had to derive these myself at one point in my academic career. It made my eyes bleed

Oh come on! The derivation isn’t that bad. Secondly, modified duration=McCauley’s Duration multiplied by 1/(1+y). @krokodilizm- duration is predominantly used to measure price sensitivity to interest rate changes. But the formula’s calculation output is in ‘years’ for whatever it’s worth.

Yea my mistake - 1/(1+y). My brain is fried at this point.

Not that they are going to test this but the output to macaulay duration comes out to match the periodicity so if you are doing a semiannual pay bond you would divide by 2 to get the years. Gotten that mixed up a few times as well - I crunch the numbers and think how the **** can the duration of a 10 year bond be 14???

Haha… That’s why the name bfry??