interest rate futures contracts used to duration hedge a fixed-income position

[JulioFRM]

Hello, where can I find the explanation to answer questions related to: compute the number of interest rate futures contracts used to duration hedge a fixed-income position. If you are given two durations, you do NOT want to hedge with the current durations, but RATHER the expected forward durations at maturity.

Thanks a lot.

 

[David Harper CFA FRM]

Hi @JulioFRM The nearest to an explain of which I am aware is simply the below in Hull (it isn't an explanation, sorry). I've previously wondered why an average isn't superior, or rather, I tended to agree with @PL that "from [his] point of view it would be preferable a duration between the duration at the beginning and at expiration of the hedging period." (see https://forum.bionicturtle.com/thre...e-used-for-hedge-calculation.5858/#post-17021). Although the logic of "forward duration" I suppose is simple enough: Say you are long an (underlying exposure) bond position with interest rate risk and today, let's say, the duration is 3.0 years but you are concerned about the risk over the next one year. Perhaps you are planning to exit the position or at least mark the gain. Keep in mind that the maturity of a interest rate future/forward derivative contract is actually the beginning of the interest rate that is providing the hedge; e.g., if the hedge is a T-bond futures contract, maturity of the hedge (derivative contract) is when the CTD bond is delivered and this bond's price is a function of the then-prevailing rate. Just like a Eurodollar futures contract has a maturity such that the maturity is the beginning of then-prevailing three-month rate (I've noticed there is often confusion about this: folks sometimes think that maturity of interest rate derivative contract is the end of the underlying rate period, but it's the beginning of a then-prevailing rate). So, continuing, let's say we go forward in time one year and the duration somehow spiked to 4.0 years. Well, at that point in time, I think you'd prefer your hedge to operate at the future sensitivity rather than the old sensitivity, is how I might think about this. The formula (the practice questions) assume you know both future durations, of the underlying exposure and the hedging instrument, so at least these match, right? If we are correct about the future durations, then at least our hedge will approximation work one year forward. I imagine there is academic treatment on this question, I just haven't read it, thanks!

"6.4 DURATION-BASED HEDGING STRATEGIES USING FUTURES

We discussed duration in Section 4.10. Interest rate futures can be used to hedge the yield on a bond portfolio at a future time. Define:

• V(F): Contract price for one interest rate futures contract
• D(F): Duration of the asset underlying the futures contract at the maturity of the futures contract
• P: Forward value of the portfolio being hedged at the maturity of the hedge (in practice, this is usually assumed to be the same as the value of the portfolio today)
• D(P): Duration of the portfolio at the maturity of the hedge." -- Hull, John C.. Options, Futures, and Other Derivatives (Page 148).

 

[JulioFRM]

David thanks a lot for your answer.
I reviewed the question set exercises P1.T3.173 related to duration hedging, but I didn’t find an exercise that puts into practice the forward duration concept, as the calculations are done with the direct durations already given by the question, as you mentioned in your answer. Do you know where can I find a question that puts into practice this concept? Many thanks.

 

[David Harper CFA FRM]

Hi @JulioFRM Yes, right. We follow Hull's lead here: we don't ask any questions (to my knowledge) that require you to calculate an estimate of the forward duration. Rather, the estimated duration in the future is simply given as an assumption in the question. Just like any of Hull's to my knowledge; e.g., EOC 6.7:

6.7. It is January 30. You are managing a bond portfolio worth $6 million. The duration of the portfolio in 6 months will be 8.2 years. The September Treasury bond futures price is currently 108-15, and the cheapest-to-deliver bond will have a duration of 7.6 years in September. How should you hedge against changes in interest rates over the next 6 months?

... so, no we don't have any questions which embed the calculation of the forward duration, nor am I aware (off the top of my head) of where such questions are written. Thanks,

 

[Flashback]

Hi David (or someone else)

First I'm sorry if this question already has been asked, and I'm quite sure it was.

Why they discount bonds with continuous rate instead use of discrete one. I mean f. ex for Zero coupon bond $102,269 =( $100 + $ 6.125/2) x e -z1/2 instead $102,269 =( $100 + ($ 6.125/2) /(1+r/2)?

Thanks

 

[David Harper CFA FRM]

Hi @Flashback I don't know which question to which you refer but in general the problem needs to specify the compound frequency (alternatively the instrument implicitly dictates; e.g., Eurodollar futures reference a 90-day interest rate so naturally their compound frequency is quarterly). As below, the use of exp(-r*T) is the convergence as k = 1|4|12|etc increases, so the continuous rate can be viewed as a rate with a very high compound frequency, k --> ∞ where k is the number of periods per year. John Hull tends to use continuous compounding throughout his text.

In your example, probably the continuous rate was used because the problem provided spot rates in that frequency. For example, if the problem says "the six month spot rate is 4.0% with continuous compounding," the exp(-0.5*0.040) should be used!

See also https://forum.bionicturtle.com/thre...fundamentals-hull-chapter-4.10570/#post-58311

Hi @Alvaro G These formulas, that relate the discount factor (df) to the spot rate, are found in Tuckman, but the key is to realize that a discount factor, by definition, is the present value of $1.00 received in the future. Here we have a semi-annual compound frequency and a continuous compound frequency:

• df*(1+r/2)^(N*2) = $1.00 such that (1+r/2)^(N*2) = 1.0/df --> (1+r/2) = (1.0/df)^(1/[N*2]) --> r = [(1.0/df)^[1/(N*2)] - 1]*2; so we are solving for the semi-annual rate, r, in 0.84*(1+r/2)^(2*6) = $1.00
• df*exp(rN) = $1.00, or df = $1.00*exp(-rN) and taking ln(.) of both sides --> ln(df) = -r*N --> r = -1/N*ln(df); so we are solving for the continuous rate, r, in 0.84*exp(r*N) = $1.00, or equivalently 0.84 = $1.00*exp(-r*N). I hope that helps!

 

[Flashback]

Yep. And Rc is used in BSM as well. So far I used to calculate with discrete rates bonds yield. I referred on Schweser IR chapter 3rd book P1.