Quick Quiz on CTD bond & futures (highly testable)

CFA Level III
#1

I’m confused about this question as to why the answer is the given one. Take a look for yourself!

Reduce the duration of the bond portfolio from the current duration of 4. The portfolio market value is currently $20 million. The futures contract is priced at $100,000. The duration of the cheapest-to-deliver (CTD) bond is 5 years. The conversion factor for the futures contract is 1.2. Question: How many futures contracts should you sell in order to reduce the duration of the portfolio to 2?

PLEASE EXPLAIN YOUR ANSWER.

A. 80

B.88

C. 96

(Scroll down to see answer.)

(The correct answer is C… If you chose C, please explain why you didn’t choose A. In other words, why did you multiply by 1.2 when the price given is the futures price, not the CTD bond price.)

#2

because CTD duration is not the same as futures contract duration. here, the futures contract duration is 5 divided by 1.2.

#3

DD futures contract = DD ( cheapest to deliver) / conversion factor

#4

(2-4) / (5/1.2) * (20m/100) = -96 or (2-4) / 5 * (20m/100) * 1.2 =-96

as you can see you can either multibly or divide the covrsion factor. dividing the denominater by a number is the same as multiblaying the numerator

#5

You have to understand the above before you realise the below formula

(Duration target - Duration portfolio) X Portfolio Value Number of contracts = ----------------------------------------------------------------------- X conv factor X yield beta Duration of Futures x Futures Price Which is why conv factor is in the numerator. Otherwise just remember to formula.

#6

you guys (and it looks like the question is too) are making the implicit assumption that the price of the futures contract is the same as the price of the CTD bond.

the following two equations produce the same result:

(DDtarget - DDp) / DDf and [(Dtarget - Dp)*Pp / DCTD*PCTD] * CTD conversion factor

you can even throw a yield beta in there if you want. it’s irrelevant.

for these equations to be equal (assuming numerators are equal):

DDf = Duration f * price f = Duration CTD * price CTD / conversion factor.

in order to solve for DDf we need the price of the CTD. we don’t have this price. if you want to see an example of what I’m saying view Level III PM 2013 mock exam from CFAI. question 50. if you simply say the Duration of the futures = duration of the CTD / conversion factor and try and solve using this duration and the futures price you won’t be correct. this is only the case when the prices of the two are equal

basically with the info given here you can’t calc the dollar duration of the futures unless you make the assumption I mentioned. this is an important point that CFAI could throw in a question. it seems like a simple shortcut to do the simple division instead of the full dollar duration calc but it won’t always lead to the correct answer

#7

guys all the formulas above are the same and will lead to the same answer, please correct me if im wrong

#8

The CTD multiplier doesn’t adjust for duration; it adjusts for market price.

#9

After going through this all over again i fully understood michaelwcao problem much better. Based on the way we calculate:

[(Dtarget - Dp)*Pp / DCTD*PCTD] * CTD conversion factor

(2-4)*20m / (100k * 5) * 1.2 = 96

There seems to be this implicit understanding that “The futures contract is priced at $100,000.” refers to Pctd = $100. My understanding of the conversion factor is as follows:

Pctd = Pf * conv factor or DDctd = DDf * conv factor

So should it be Pf = $100,000 or Pctd = $100,000? And what would be DDctd and DDf?

Getting confused

#10

you all are really overcomplicating this. CBOE gives you the conversion factor. so just take (target dur - port dur) / dur of ctd multiply by (port value / price of ctd).

The future value only comes into play if CTD not mentioned at all or if you need to calc it.

conversion factor = (duration of CTD * price of CTD) / dollar duration of futures contract