# Hedging bond with futures: yield beta

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Why do we ever need to assume that the yield spread between a hedged bond and the futures contract’s cheapest-to-deliver bond is constant?

It only makes sense to me that they should ideally have the same duration to eliminate portfolio’s interest rate risk. Duration is important, yes. But how does a constant yield spread come into the picture?

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Including Yield Beta is essential for hedging.

Consider the following example.

You are holding \$1,000 worth of Corporate Bond A and you are trying to hedge this Corp bond A with  CTD Treasury futures with a denomination of \$1,000.  (Futures for your Corporate Bond A do not trade on the market and hence you have to hedge with Treasury futures).  The question is how many Treasury futures should you be selling?   The answer is derived as follows.

You have run the following regression on historical yields.

Corporate Bond A’s yield = alpha + ( beta * CTD Treasury future’s yield ) + error.

In this regression, beta turns out to be 2, let us say.  That is, for every 100 basis point move in CTD Treasury futures, your corporate bond A moves 200 basis points.

Assume that after you hedge by selling treasury futures, the treasury future’s yield changes by 200 bps and consequently your Corp Bond’s yield changes by 400 bps.

Thus to perfectly hedge your \$1,000 worth of corp bond A, you should sell two \$1,000 worth of CTD Treasury futures, for a total of \$2,000.

Thus, to get the number of futures, you have to multiply by yield beta.

Hope this response helps in clearing up your doubt.

P

Best Regards

P

Hi psriniva. In your example, yield on corp bond increases by 400bps, and yield on T bond drops by 200bps. But yields are not additive, are they? The price effects of T bond yield moving down 200bps and corp bond yield moving up 400bps don’t exactly cancel, I guess. That’s why I think dollar duration is more relevant.

Really confused.

1. using futures you are hedging your position (offsetting gains / losses)

2. It is a tactical allocation using futures contracts - which gives you the following

b. you use futures contracts that are less expensive for a short period of time based on expectations to shift your position.

dollar duration makes sense if you were selling something and buying something else.

CP

No yields are not additive.  I did not imply that at all.  If a corp bond’s yield beta is 2, it kind of means that the corp bond is twice as volatile as the Treasury.  This is similar to a stock having a beta of 2 compared to S&P 500.   Say that the Fed is selling bonds in the open market and hence the market interest rate is going up and now if the treasury’s yield goes up by 100 bps, the corp bond A’s yield may go up by 200 bps).

In addition to what CPK123 has said, the following example may illustrate as to why considering dollar duration is not sufficient and yield beta needs to be considered in determining the number of futures contracts.

Let us assume you have invested \$1,000 (Principal) in Corporate Bond A.  You want to protect this investment.

You are going to sell Treasury futures each with a Notional Principal of \$1,000.

Let us assume initially that the duration of treasury futures is 5 and is same as the duration of the Corp. Bond A. (Also 5).

Corp. Bond A has an yield beta of 2 meaning that for every 100 bps move up in Treasury futures, the Corp Bond A’s yield will move up 200 bps or for every 100 bps move down in Treasury futures, the Corp Bond A’s yield will move down 200 bps.  (Based on the regression of historical yields - see my earlier post).

Now let us say that the yield on Treasury futures moved up 100 bps.  That means the price went down by 5 %, ie. \$50 (dollar duration).  By the way, the Corp. Bond As’ dollar duration is also \$50 only.

Meanwhile, due to yield beta being 2, the corp Bond A’s yield has gone up by 200 bps.  The duration being 5, this means that the Corp Bond’s price has gone down by 10 % (2 times 5 because of 200 bps move).  That is the corp bond A’s price has gone down \$100.

By considering only dollar duration, if you had sold only one Treasury future as a hedge, you would gave gained \$50 with the Treasury futures and would have lost \$100 in the Corp Bond A’s investment.  So, you have under hedged by not considering yield beta and by just considering dollar duration.

By including yield beta in your consideration, you would have sold 2 Treasury futures and you could have gained \$100 (\$50 each times 2).  This \$100 gain would have perfectly offset the \$100 loss with Corp Bond A.

So, the bottomline is that whenever we are hedging one investment such as a Corp Bond with a different insturment such as Treasury, we need to include yield beta between the two instruments in determining the number of futures contracts.

Finally, if we had been trying to hedge “On the run” Treasury Bond investment with Treasury futures, yield beta is unimportant because it is either 1 or close to 1.

Hope this clarifies.

P

Best Regards

P

I think I got it now, psriniva. Your note was helpful. Am I right to say the following: it’s “dollar duration x yield change” that actually matters. But if that’s true, why don’t we hedge by matching the product of dollar duration and yield beta of the CTD with that of the hedged bond? Do we really have to match both dollar duration and yield beta (looks to me this is a suffcient, but not necessary, condition for perfect hedging)??

you never match dollar duration.

you just find the # of futures contract using the yield beta approach.

CP

How is that true, cpk123? Matching yield beta alone can hedge your bond against interest rate risk? The date you lift your hedge, value change in your futures should equal value change in your bond. To make that happen, price change = dollar duration times yield change must be equal for the hedged bond and the CTD. Is it not so?

Kazec,

Yes, I agree with you that dollar duration does not necessarily have to be matched.  In my example, I matched the dollar duration of the corp bond A and treasury only to make the example easier to illustrate the effect of yield beta.

It is the combination of dollar duration, yield beta and the Notional Principal,  that will determine the number of futures to sell.

P

Best Regards

P

there are two pieces to this whole thing.

1. How many Treasury futures to buy/sell - to make sure you meet the Duration expectation (Target duration). Over there - the duration (current, expected and futures), value of portfolio, value of Futures contract and the yield beta play a role.

2. This is a tactical allocation - since you are NOT selling / buying your original portfolio.

At the end - you evaluate whether your portfolio met the desired. In the curriculum - they do this only for the equity portfolio - where they calculate an effective beta - but do not show any similar calculation for the target duration achieved - especially since the duration is a much more nebulous concept.

CP

Thanks P, for the confirmation.

Sorry CP, still don’t know what you mean.

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