# Dollar Duration/Conversion Factor Hedge Question

Ok quick question in Schweser vol I exam III they calc dd two different ways per the below: One is incorporating the conversion factor and one is not and when they calc the # of contracts they divide by conversion factor for both. Why are they including the conversion factor in 18.4 (the 1.259)? 18.4 The dollar duration of the CTD for a change in YTM of 50 bp based on the expected price at expiration is: 108,500 × 1.259 × 0.005 × 10.15 = 6,932.53. 15.1 For the CTD, the dollar durationCTD = 9.48 × 0.008 × 139.72 = \$10.5964 per \$100 of par

I dont have Schweser but I’m assuming the 108,500 is the price of the CTD which needs to be adjusted by the Conversion factor to equate it to a 6% 20Y Treasury. Then the rest is the same. I’m guessing the 139.72 in the second problem was already adjusted.

This is a mistake on their part, you don’t need to use the CTD conversion factor if the future contract given is based on the CTD bond. They actually used the CTD conversion factor twice in the same calculation in this problem (if I remember).

all i know whnever they say there is a future with duration bla bla you have to use CTD factor, it is even in one of the blue examples in CFAI. They say Future duration is bla bla but really mean it is CTD duration

MO: i am pretty sure even if the futures is for the CTD bond you still have to use the CTD conversion factor…unless, someone tells me i’m wrong…

I’m not sure about the “Blue Box” … I adjust the hedge ration (DD port/ DD future) for the fact that the future contcact I will be using in the real world will be based on the CTD bond.

3rd & Long Wrote: ------------------------------------------------------- > MO: i am pretty sure even if the futures is for > the CTD bond you still have to use the CTD > conversion factor…unless, someone tells me i’m > wrong… Hedge ratio = DD portfolio / DD Future = (DD Port / DD CTD ) * ( DD CTD / DD Future) = (DD Port / DD CTD) * CTD_Conversion_Factor. As you can see if the future given is based on the CTD bond, then the ratio DD_CTD/DD_F = 1

let’s not get off topic i’m talking about when calculating the dd schweser seemed to use the ctd conversion factor, I know we always need to divide by it in the dem. of the equation to get to # of contracts unless stated otherwise.

Search “CTD” and you’ll find a lot of debate about this. I firmly believe that if you’re given the duration and price of the CTD, then you use the conversion factor; but if you’re given the duration and price of the futures contract, you do not. However, CFAI’s examples, end of reading problems and free sample exam use the conversion factor even when you’re given the price and duration of the futures. This has been reported as errata to CFAI at least twice to no avail.

What’s this “futures based on the CTD bond”? I guess it would be nice if there were futures contracts in which only one bond was deliverable in which case there would be no need for any of this conversion factor stuff. Alas, it is not the case. The only bonds that have conversion factor = 1 are bonds with 6% coupons (for US bond contracts). So this stuff: “(DD Port / DD CTD) * CTD_Conversion_Factor. As you can see if the future given is based on the CTD bond, then the ratio DD_CTD/DD_F = 1” is only true if the CTD is a 6% coupon bond.

I’ll take a look but your talking about the calc on dd on the hedging instrument?

So just thinking about this for a second - one of the problems with these hedging things is that there is a big mismatch in hedging bonds with futures contracts. Suppose you have \$100M face value of bonds with CF = 1.2. To hedge interest rate movements in those, you are going to short something like 1200 contracts. At expiration, if you deliver up your bonds on the futures contracts you still have a short position of -200 bond contracts (this is called “the tail”). That’s unlike any commodity where if you have 10000 oz of gold and you short 100 futures contracts you are earning rf + carry and deliver up all your gold on the contract and you’re done. It might be that the hedging confusion comes from that problem.

s23Dino – I should have refreshed my memory about those questions before posting. Is the confusion that in 18.4 you’re given the price of the futures contract, but the duration of the CTD, while in 15.1 you’re given everything for the CTD? In 18.4, to get to the dollar duration of the hedging instrument, are they doing the following? Pctd = Pf x CF DDctd = Pctd x DURctd x .005 DDf = DDctd/CF The number of contracts then equals: -DDp/DDf x Yield Beta Don’t know if that’s right, but looking at the two questions I seem to remember something along those lines…

For 18.4 it seems to calc dd of the ctd they used: 18.4 The dollar duration of the CTD for a change in YTM of 50 bp based on the expected price at expiration is: 108,500 × 1.259 × 0.005 × 10.15 = 6,932.53. and the 1.259 is the conversion factor. I think mo34 above was on the right track they are using the conversion factor twice, once to calc dd of ctd future and then they are dividing it in the equation to get to the # of contracts used to hedge.

JoeyDVivre Wrote: ------------------------------------------------------- > What’s this “futures based on the CTD bond”? I > guess it would be nice if there were futures > contracts in which only one bond was deliverable > in which case there would be no need for any of > this conversion factor stuff. Alas, it is not the > case. The only bonds that have conversion factor > = 1 are bonds with 6% coupons (for US bond > contracts). So this stuff: > > “(DD Port / DD CTD) * CTD_Conversion_Factor. > > As you can see if the future given is based on the > CTD bond, then the ratio DD_CTD/DD_F = 1” > > is only true if the CTD is a 6% coupon bond. Agree. The wording of the problem mentioned " Duration of the Cheapest to Deliver Future contract" . This implies that the conversion factor is already taken care of.

MaxTheDog Wrote: ------------------------------------------------------- > s23Dino – I should have refreshed my memory about > those questions before posting. > Is the confusion that in 18.4 you’re given the > price of the futures contract, but the duration of > the CTD, while in 15.1 you’re given everything for > the CTD? > > In 18.4, to get to the dollar duration of the > hedging instrument, are they doing the following? > > Pctd = Pf x CF > DDctd = Pctd x DURctd x .005 > DDf = DDctd/CF > > The number of contracts then equals: > > -DDp/DDf x Yield Beta > > Don’t know if that’s right, but looking at the two > questions I seem to remember something along those > lines… Exactly, if the information given is for the CTD there is no conversion needed. That’s my point.

all i know that in sample test they said future duration is bla, and then convert it using CTD, and the same thing in their example in the book.

I don’t understand your question then dino. To me the approach in the two answers seems consistent. The difference is that in 18.4 you’re given the price of the futures contract, but in 15.1 you’re given the price of the CTD. What am I missing?

mo34 Wrote: ------------------------------------------------------- > Agree. The wording of the problem mentioned " > Duration of the Cheapest to Deliver Future > contract" . Did it really? That just doesn’t make sense. There’s a futures contract and a CTD bond but not a CTD futures.

No, question 18 gives “futures contract (\$100,000 face) currently priced at 108.5” and “CTD bond…duration of 10.15”. Question 15 gives “price of the cheapest to deliver bond…conversion factor…and duration” (all of the CTD).