q: Observing the 6 month futures price, an investor concludes that the CTD treasury bond has an expected DD of $6954 for a 75bps change in interest rates six months from today. Using this, he concludes that a corporate bond he holds has an expected DD of $8.559 per $100 for a 75bps change in interst rates six months from today. The value of his holding is $10mln. The conversion factor for the CTD bond is 1.156. If he wants to hedge against a possible rise in rates of 75bps, the manager should: a) sell 94 contracts b) buy 94 contracts c) sell 142 contracts d) buy 142 contracts
DD(F) = DD(CTD)/Conv factor
A B C D?
I don’t remember the formula so will start from first principles: To hedge completely you want the dollar duration of the target portfolio to be zero. Since duration of your long position is positive, you will need negative duration from futures. That tells you that you need to sell futures. -duration of target portfolio = -duration of current portfolio + -duration of futures 0 = (8.559 X 10^7)/100 + Nf X -duration per future 0 = (8.559 X 10^7)/100 + Nf X -duration of CTD/1.156 0 = (8.559 X 10^7)/100 + Nf X 6954/1.156 ==\> Nf = -(8.559 X 10^7)/100 X (1.156/6954) = -142.28 futures Sell 142 futures. I am aware that -duration is calculated for 100 bps change in interest rates; in this case, this immaterial as 100/75 factors cancels out through the equation. OK, this was too easy… so I must have f… up. Is the answer sell 94 contracts?
hedge a holding, i have to sell, a) or c) since DD of bond >> DD of futures, hedge ratio > 1. so c) … right?
ok - thanks. i couldn’t figure out why 8.559 was divided by 100 and i knew i was overlooking something stupid. I overlooked the 1bps part of it. That’s why its divided by 100. ya, the answer is c) sell 142 contracts. Thanks guys.
it doesnt matter if it is 75bps or 2000bps, as long as Dollar duration of both instruments is measured using the same IR curve shifts. Again, bigger the shift worse the hedge
comp sci- I, like you, missed the ‘trick of the question’ …thetrick had nothing to do with the 75bps but the fact taht the dollar duration was given per 100 and not for a per bps shift.
strikershank Wrote: ------------------------------------------------------- > comp sci- I, like you, missed the ‘trick of the > question’ …thetrick had nothing to do with the > 75bps but the fact taht the dollar duration was > given per 100 and not for a per bps shift. not sure i understand you. if you really want to get DD for the 100 bps change, you need to divide the given DD by 75, then multiply by 100. right?
rand0m Wrote: ------------------------------------------------------- > strikershank Wrote: > -------------------------------------------------- > ----- > > comp sci- I, like you, missed the ‘trick of the > > question’ …thetrick had nothing to do with > the > > 75bps but the fact taht the dollar duration was > > given per 100 and not for a per bps shift. > > not sure i understand you. if you really want to > get DD for the 100 bps change, you need to divide > the given DD by 75, then multiply by 100. right? Yes. i don’t understand striker’s point either.
The solution reads (which i didn’t get at first until this thread) A: ([-(8.559/100)*$10mln] / $6954)*1.156 = -142. so my interpretation is that because they gave dollar duration per $100 as opposed to 1bpg you have to divid the DD by 100 (or multiple it by 0.01)…
my translation to that piece of english is the DD of 8.559 is calculated based on 75bps changes for each $100. the reason they need to divide by 100 is to find out how many $100’s in 10 mln (or corresponding DD for each dollar). i guess this is really a 5th grade math problem, instead of a fixed-income problem … 
Agree with random. The idea is to find DD for 75 bps of $1 face value. when we multiply by 10 million we get the total DD for the holdings.