# How to hedge a bond against 50 bps change in int rate?

Hi, I am stuck with this problem where bond futures are to be used to hedge the risk of a 50 bp change in the market rate of a 20 year bond a company will be issuing in 270 days. See the specifics below. (Ref: Schweser Exam 3, Question 18, Page 180) Q1. How does one hedge a bond against a 50 bp increase in market interest rate? I know how to change or remove the duration of a bond but don’t know how to hedge the bond against a specific movement in int rate. Q2. Why the question has provided two durations for the bond? =================== Specifics: Face value of bond: \$25 million If issued today would have an eff duration of 9.94, and has an expected eff duration at issuance of 9.90 based on a constant spread assumption Yield beta =1.05 Current spread of the company’s corp debt to Treasuries=2.4% Futures face value: \$100,000. Price: 108.5 CTD conv factor:1.259, Duration: 10.15, implied yield:6.4% =================== Thanks, MG.

9.90 x 0.50% x Market Value of Bond = Dollar Duration That’s the amount you would need to hedge or in other words, your target dollar duration. YOu didn’t include the market price of the bond so unless I’m missing something here…

Ahh… I see it was just issued so it should be: \$25M x 9.90 x 0.50% = \$1,237,500 Dollar Duration Target DD of CTD = 10.15 x 1.259 x 0.50% x \$100,000 x 1.085 = \$6932.53 So: (\$1,237,500 - 0)/\$6,932.50 * 1.05 = 187 Contracts??? Not sure I did this right…

PJStyles Wrote: ------------------------------------------------------- > Ahh… I see it was just issued so it should be: > > \$25M x 9.90 x 0.50% = \$1,237,500 Dollar Duration > Target > > DD of CTD = 10.15 x 1.259 x 0.50% x \$100,000 x > 1.085 = \$6932.53 > > So: > > (\$1,237,500 - 0)/\$6,932.50 * 1.05 = 187 > Contracts??? > > Not sure I did this right… you did but I think on the exam your calculation would be wrong (even though its technically correct) because CFAI interchanges duration of future for the duration of CTD

> you did but I think on the exam your calculation > would be wrong (even though its technically > correct) because CFAI interchanges duration of > future for the duration of CTD What do you mean exactly? Can you expand… would we just use the CTD duration without adjusting it?

I got this one wrong as well, should you not divide the conversion factor to get CTD DD?

I must also state that the 1st time I saw this, I divided by the conversion factor. Not sure I understand when to divide by the conversion factor and when to multiply… i just don’t get it.

PJStyles Wrote: ------------------------------------------------------- > I must also state that the 1st time I saw this, I > divided by the conversion factor. Not sure I > understand when to divide by the conversion factor > and when to multiply… i just don’t get it. dividing by the conversion factor is the same as multiplying it. ie 10/(1/10) = (10/1) * 10 this was discussed before, in sample exam 1 cfai solved for DDctd by multiplying price of future by its dollar duration while schweser gets DDctd with price x duration of future x conversion factor. search the topic and you’ll get a better sense of it.

you totally lost me just now… crap… gotta go find this thread you speak of… grrr…

I didn’t get to worried about this one - I think I’m just resigned this to being one of the questions that you’re not really expected to get correct.

• dollar duration of your bond = 25,000,000 x 9.90 x 50 / 100 = 1,237,500 + num futures = (DDportfolio / DDfutures) x yield beta + you can use the ctd instead of the futures, taking into account that + DDfutures = DDctd / F + num futures = (DDportfolio / [DDctd / F]) x yield beta + num futures = (DDportfolio / DDctd) x yield beta x F The problem with this example is that you don´t know DDctd. They are telling you the price of the futures but the duration of the ctd + DDctd = price of ctd x nominal x ctd duration x change in yield If you assume that price of ctd = price of the future x F + DDctd = price of future x F x nominal x ctd duration x change in yield + DDctd = 108.5% x 1.259 x 100,000 x 10.15 x 50 / 100 = 6,932.53 + num futures = (DDportfolio / DDctd) x yield beta x F = (1,237,500 / 6,932.53) x 1.05 x 1.259 = 236 For me, the following works: 1. When using ctd, remember to have DDctd in the denominator and multiply by F (and by yield beta). 2. When ctd is not mentioned, use the “good one” of (MDURt - MDURc)/MDURf times mkt value of the bonds divided by futures price. This is easier because it works equal to the one of equities (using MDUR instead of beta). By the way, CFA texts always uses yield beta = 1 in this ones Regarding this schweser question, is too tricky, I am sure we will not face this on saturday. If they give you the ctd data, they will tell you both ctd duration and F. If not, just MDUR of the futures and futures price. They will not mix them in such a tricky way hope it helps

Thanks a lot folks.

hala_madrid, Thanks for your detail explanation. 1. I was confused because I thought: Duration of futures contract = Duration of CTD / Conversion factor …which is not correct. The conversion factor applies only in the case of Dollar Duration and not in the case of plain Duration. i.e. DD of futures contract = DD of CTD / Conversion factor. (Ref: CFAI book Vol 4, page 17.) 2. Also you made an important assumption which seems correct in this case but not correct in practice. Price of CTD = Price of futures contract x Conversion factor. Conversion factor is a function of the bond’s coupon rate and not a function of its current mkt price. (Ref: slide 5 of this file: http://sbufaculty.tcu.edu/mann/Fin7523_s99_15_TBonds_hedging.PDF ) Thanks again, MG.