How do you calculate the yield of the Treasury bond futures contract if you know the price? -- Gene Kwek
The Treasury bond futures contract traded on the
Chicago Board of Trade
is a futures contract, not a bond, so it doesn't have a yield like a bond, but I understand what you're getting at. The Treasury futures contract trades in lockstep with the 30-year Treasury bond itself (often called the cash bond, to distinguish it from the futures), such that a given futures price seems to correspond to a cash bond yield. And indeed it does.
Unfortunately, the relationship is complicated. With the help of Keith Schap at the CBOT, I'll try to put it in simple terms.
The Chicago Board of Trade
lists four Treasury futures contracts -- a bond contract, a 10-year note contract, a five-year note contract and a two-year note contract. The bond contract has been the world's most actively traded futures contract for the last two years,
according to the
Futures Industry Association
The CBOT's listings page has more detailed information on the individual contracts, but the bond futures contract works as follows. When you buy one, you are agreeing to take delivery at expiration of $100,000 face value of Treasury bonds, or 100 bonds each with $1,000 face value. Virtually all bond contracts are cash-settled, meaning the holder takes the value of the contract in cash at expiration rather than taking delivery of the underlying bonds.
Still, it's important to understand that the futures seller decides which bonds to deliver and when to deliver them, within guidelines established by the CBOT. Those
guidelines instruct sellers to deliver, by the last business day of the delivery month, "U.S. Treasury bonds that, if callable, are not callable for at least 15 years from the first day of the delivery month or, if not callable, have a maturity of at least 15 years from the first day of the delivery month." The bond futures contract, while it trades in tandem with the newest 30-year Treasury bond, doesn't require delivery of that bond.
There are four delivery months for bond futures -- March, June, September and December -- meaning that contracts are listed for those four months. At any given time, the contract for the nearest future month, or the front-month contract, is the most active. The CBOT's
daily settlement page lists the bond futures contracts currently trading with the front-month contract first. On the
intraday quote page, the front-month contract is the one furthest to the left.
Futures-Treasury Bond Link
So what's the relationship everyone talks about between the futures price and the yield of the current 30-year Treasury bond, the benchmark bond?
It begins with the fact that the futures seller decides which bonds, in the universe of bonds fitting the CBOT's description, to deliver against the contract. Even though most futures holders don't actually take delivery, like any futures contract, the Treasury futures contract tracks the price of the underlying commodity. And like any futures contract, that gives the seller the choice of which commodity to deliver, the Treasury futures contract tracks the price of what's called the "cheapest to deliver," or the CTD bond.
I'm not going to explain what makes a bond the cheapest to deliver. It's very complicated, and it doesn't really matter. For these limited purposes, all you really need to know is that the cheapest-to-deliver bond against the Treasury futures contract is, and has been for a while, the 11.25% coupon bond due Feb. 15, 2015. At yesterday's close, it was priced 149 4/32, or 149.125, to yield 6.27%.
The value of the futures contract is based on the supposition that the holder is going to take delivery of the CTD bond. What price would the holder pay for that bond? Hint: Not the market price. If sellers could deliver any bond fitting the guidelines at its market price, they would deliver the most expensive one. So to equalize the prices of the deliverable bonds, the CBOT has developed an elaborate factor system. As its
table shows, if you want to deliver the Feb. 15, 2015, 11.25%-coupon bond against the September futures contract, the factor (which changes every contract month, or every March, June, September and December), is 1.2832.
Crunching the Numbers
That brings us to the equation that states the relationship between the cash bond price and the futures price. It is as follows:
cash price = (futures price * conversion factor) + basis
The basis, you can see from the equation, is the premium an investor would pay for the cash bond vs. the futures contract. Why would an investor pay more for the cash bond than for the contract? Because the cash bond is a bond -- it pays income. The futures contract is just a contract. It doesn't pay anything.
Unfortunately, it's slightly more complicated than that. The basis represents not just the coupon income that a holder of the cash bond would earn until delivery. It represents the difference between the coupon income and what it costs the investor to finance the cash bond position at an overnight interest rate. That difference is called the cost of carry. Cost of carry accounts for most of the basis. The rest represents the value of the delivery options to the futures seller.
Using the example above, you can see that the current basis for the September contract (the basis changes as the delivery date approaches, since cost of carry declines with time) is about 28/32. The futures settled last night at 115 17/32, or 115.531.
149.125 = (115.531 * 1.2832) + basis
149.125 = 148.249 + basis
149.125 - 148.249 = basis
basis = 0.876 (or 28/32)
So if you know the futures price and the price and yield of the CTD bond, you can figure out how much the bond's price (and therefore its yield) will change if the futures price changes by a certain amount. The problem is, unless you're paying hundreds of dollars a month for a bond market data service, you probably don't have access to the daily price of the CTD bond.
Fortunately, there's a way around the problem.
Dollar Value of a Basis Point
You don't really need to know what yield on the CTD bond corresponds to the future price. Just assume that the current futures price corresponds to the benchmark 30-year bond's yield. What you're interested in is how much the bond's yield will change if the futures price changes by a certain amount, or how much the futures price will change if the bond's yield changes by a certain amount. Here's how you do that:
For any bond you can calculate the "dollar value of a basis point," and while that figure will change slightly as the bond ages and its price changes, for the CTD bond, the figure is in the neighborhood of 13 cents. Meaning that for every basis point change in the bond's yield, its price will change (moving in the opposite direction as the yield) by 13 cents per $100 of par value.
You can also calculate the dollar value of a basis point (the DV01) for the Treasury futures contract, by taking the DV01 for the CTD bond, and dividing it by the appropriate CBOT factor. In this case it's:
0.13 / 1.2832 = 0.1013
So for each basis-point change in the yield of the CTD bond, the futures contract will either gain or lose $0.1013, or about 3/32 in price terms.
The equation is:
yield change in basis points * DV01 = price change
For example, a five-basis-point yield change should produce a price change of 0.507, or 16/32.
5 * 0.1013 = 0.507
To calculate how big a yield change would correspond to a given price change, you can turn it around like this:
price change / DV01 = yield change
For example, an 8/32 (0.25) change in the price of the futures contract should produce a yield change of about 2.5 basis points.
0.25 / 0.1013 = 2.47
The yield change indicated by this exercise is for the CTD bond rather than for the benchmark 30-year issue, but the assumption is that if the CTD's yield changes by a certain amount, the benchmark issue's yield will change by a like amount. For rough estimates, it's a safe assumption.
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