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I am interested in how the price of the ZN contract is calculated using the 10yr note yield. I am sure it has been explained on FIO before. Could anyone point me toward a thread or article or just explain how to calculate the ZN price?
Thanks,
Blackgrey45
Can you help answer these questions from other members on NexusFi?
AI is good for these types of questions. This is what Google Bard says (free subscription, better than ChatGPT for a lot of financial stuff as the latter was last updated in 2021):
The price of the ZN futures contract is calculated using the 10-year note yield as follows:
Price = 100 * exp(-yield * (365 / 100))
where:
Price is the price of the ZN futures contract in dollars
Yield is the 10-year note yield in decimal form
365 is the number of days in a year
100 is the face value of the ZN futures contract
For example, if the 10-year note yield is 2.5%, then the price of the ZN futures contract would be:
Price = 100 * exp(-2.5 * (365 / 100)) = 100 * 0.907047 = 90.7047
The price of the ZN futures contract is inversely proportional to the 10-year note yield. This means that as the 10-year note yield decreases, the price of the ZN futures contract will increase. Conversely, as the 10-year note yield increases, the price of the ZN futures contract will decrease.
The ZN futures contract is a popular hedging tool for investors who are exposed to interest rate risk. By buying ZN futures contracts, investors can lock in a fixed interest rate for a future period of time. This can protect them from the risk of rising interest rates, which could lower the value of their investments.
Here is an example of how an investor could use ZN futures contracts to hedge against rising interest rates. Let's say an investor owns a bond that matures in 10 years. The bond is currently trading at par, but the investor is concerned that interest rates will rise in the future. If interest rates do rise, the value of the bond will fall. To hedge against this risk, the investor could buy ZN futures contracts. By doing this, the investor would be locking in a fixed interest rate for the next 10 years. This would protect them from the risk of rising interest rates, which could lower the value of their bond.
You can't outsource confidence in trading decisions
Should be noted that the ZN Future contract is based upon a 7-10 year Bond with a coupon of 6%. Hence if interest rates were 6% the price would be 100!
I generally find AI is terrible at very factual questions but excellent at long winded wordy answers.
The formula you provided for calculating the price of the ZN futures contract is correct, but it is not the only factor that determines the price of the contract. The price of the ZN futures contract is also influenced by other factors, such as the current price of the underlying Treasury note, the level of volatility in the market, and the expectations of traders about future interest rates.
In the example you provided, the 10-year note yield is 3.8435%. Using the formula you provided, the price of the ZN futures contract would be zero. However, the actual price of the ZN futures contract is 112.1328. This is because the other factors that influence the price of the contract, such as the current price of the underlying Treasury note and the level of volatility in the market, are offsetting the effect of the low interest rates.
The fact that the price of the ZN futures contract is not zero even though the 10-year note yield is very low suggests that traders are expecting interest rates to rise in the future. This is because if interest rates were to rise, the price of the underlying Treasury note would fall. To protect themselves from this risk, traders are buying ZN futures contracts, which will appreciate in value if interest rates rise.
The ZN futures contract is a complex instrument, and there are many factors that can influence its price. The formula you provided is a good starting point for understanding how the price of the contract is calculated, but it is important to remember that other factors also play a role.
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Bond Math as it's called is pretty heavy math and while I do know a little I'm not an expert. (I trade primarily energies).
Starting with an explanatory example. Let's assume that 1yr rates are 5% but I have a 1yr Bond I want to sell that yields 6%. What should the price of that bond be? Well we know that if we buy the 1yr 5% that in a years time we will have 105% of our initial capital. To get the same ending capital with the 1yr 6% bond, I can afford to pay 101 rather than 100 for the bond. In that case I would lose 1% of my capital when the bond is redeemed at par, but I would make 6% interest, hence I would have the same 105% of our initial capital after a year.
This gets a lot more complicated when you consider time periods other than a clean 1 year horizon, and when you take into fact that interest is paid every 6 months. Hence a 10 year bond has 20 interest payments and a return of principle. These all have to be discounted back to todays dollars using interest rates. It gets even more complicated that the interest rate for the interest payment in 6 months time is different to the one in 12 months, 18 months etc. Then add in the fact that gains on the bond can be classified as capital gains and hence taxed lower than interest and it gets really complicated.
Second (less) simple example. This is very dependent upon discounting cash flows. We know that if interest rates are 5% we invest $1, that we will have $1.05 in a years time and $1.1025 in 2 years time. Hence it follows that $1 in a years time is only worth 95.24c today, and $1 in two years time is only worth 90.7c.
Keeping with our 5% interest rates assumption our first bond is a 2yr bond that pays 5% interest every 12 months. What is this bond worth? Well in year years time we get 0.05 interest which we know is only worth 0.0476 today and in two years time we get another 0.05 interest which we know is only worth 0.0454 today. In two years time we also get our money back, but that is also only worth 0.907 today. So the bond is worth +0.0476+.0454+.907 = 1. Viola!
Now consider a second bond, a 2yr bond that pays 6% interest every 12 months. What is this bond worth? Well in year years time we get 0.06 interest which we know is only worth 0.0571 today and in two years time we get another 0.06 interest which we know is only worth 0.0544 today. In two years time we also get our money back, but that is also only worth 0.907 today. So the bond is worth +0.0571+.0544+.907 = 1.0185
Back to the question, how do you calculate the yield based upon a futures price. In reality you can't use a simple formula. If you check the CME website you will see that ZN has a "Grade of Quality" of
U.S. Treasury notes with a remaining term to maturity of at least six and a half years, but not more than seven and three quarters years, from the first day of the delivery month. The invoice price equals the futures settlement price times a conversion factor, plus accrued interest. The conversion factor is the price of the delivered note ($1 par value) to yield 6 percent.
So there isn't a single bond that can be delivered into the contract there's probably dozens. Each of those bonds has a different expiry, different coupon, and a different CME conversion factor. So you would need to take every bond that is deliverable and back into which is the cheapest to deliver into the contract. The yield of that bond is the equivalent yield of the ZN futures contract.
You will note that the deliverable bond has a maturity of at least six and a half years, but not more than seven and three quarters years. Hence ZN is really a 7 year bond future not a 10 year! If you really want to trade the 10 year you need to be trading TN the "Ultra Ten Year Bond" contract which are based upon Original issue 10-Year U.S. Treasury notes with not less than 9 years 5 months and not more than 10 years of remaining term to maturity. This didn't use to be the case. In late 2022 CME changed the deliverable maturities for ZN from 10 years to the current 7years 9months but they didn't change the name! They did this because the cheapest to deliver was often a short duration bond and not a 10 year.
This is a great point @SMCJB - same for the 30Y. The 30Y futures are 15-25 years, so, you're really looking at a 20 year bond most likely The 30Y Ultra is 25 year to maturity or greater.
As can be expected, since the treasuries to deliver are more limited, the futures are thinner. About 15% as liquid in the 10Y ultra compared to the standard, and about 50% as liquid for the 30Y ultra compared to the standard (you actually see people trading the 30Y ultra, can't say I've ever seen anyone really trade the 10Y ultra).
The 30Y Ultra is 25 year to maturity or greater.