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# Calculating futures equivalent yield - 10yrNote

Discussion in 'Financial Futures' started by futures_shark, May 26, 2005.

1. ### futures_shark

I've been trying to figure out the equivalent yield based on the futures price.

Using this formula:

where:
C = coupon payment
n = number of payments
i = interest rate, or required yield
M = value at maturity, or par value

For n I'm substituting the actual time (in years) from futures expiration until the payment date. In excel its' the (pmt date - futures exp date)/365.25 **(365.25 to account for leap yr)

I'm using the most recent auctioned note along with the conversion factor from the CBOT site.

I'm adding up all the present values of the future payments on a semiannual schedule to come up with the present value of the payments on expiration day. Then using excel to solve for the interest rate that would equate this calculation to the current futures price multiplied by the conversion factor.

This should provide a result close to \$tnx quote but it's not working out.

Is there something missing from this methodology? I've double checked the equations and everything seems to be correct...except the answer.

Code:
```5/16/2005  Issue date
4 1/8	   Interest rate on Note (\$2.0625 semiannually)
97.09	   PV Of future payments
6/21/2005  Futures Expiration
112.5	   Futures Price
0.8630	   Conversion factor
97.0875    Converted price
4.596%	   Futures Equiv. Yield

Date             PV
11/16/2005	 2.03
5/16/2006	 1.98
11/16/2006	 1.94
5/16/2007	 1.89
11/16/2007	 1.85
5/16/2008	 1.81
11/16/2008	 1.77
5/16/2009	 1.73
11/16/2009	 1.69
5/16/2010	 1.65
11/16/2010	 1.62
5/16/2011	 1.58
11/16/2011	 1.55
5/16/2012	 1.51
11/16/2012	 1.48
5/16/2013	 1.45
11/16/2013	 1.41
5/16/2014	 1.38
11/16/2014	 1.35
5/16/2015	65.41 Principal + interest

```

2. ### Aaron

Hmmm, I think you are missing a couple things. Let's work step by step...

My best guess of where the price of the cheapest-to-deliver t-note will be at expiration of the futures contract is the current cash price. And that is what I am willing to contract now to pay at that time...

F* = S

But wait, the people that actually hold the t-note might get one or more coupon payments before my futures contract expires, so I want to reduce what I am going to contract to pay for the t-note at expiration by the present value of those coupons:

F* = S - PVC

But wait, F* is what I could pay now to buy that t-note and take delivery at expiration. I don't want to actually pay for the t-note until delivery. I just want to contract to buy it now, not actually pay for it. Instead I'm going to borrow at the short term risk free rate to finance my purchase and then pay back the loan when I make my actual purchase at expiration. The amount of money I'll have to pay at expiration is

F* = (S - PVC)(1 + r)^t where t is the life of the futures contract

And then finally, let's convert F* to a standardized futures price (F) by dividing by the conversion factor for a nominal 6% yield t-note. We can get the conversion factor (CONV) from the CBOT.

F = [(S - PVC)(1 + r)^t] / CONV

See if this formula gets you any further, Futures_Shark.

Aaron Schindler

3. ### ramuk

I contacted CBOT regarding this and they sent a spreadsheet my way. But it's bigger
than the max size to attach here.

So perhaps you can also contact CBOT.

4. ### futures_shark

Aaron, I don't have the cash price so that method won't work. Plus I'm looking for the yield basd on the futures price, not the theoretical futures price.

Ramak,

Thanks for the tip, I'll check with the CBOT.

5. ### Aaron

How about emailing it to me, Ramuk, and I'll put it up on the SchindlerTrading.com website and provide a link to it here? My email address is aaron @ schindlertrading.com (without the spaces).

Aaron Schindler

6. ### ramuk

I have emailed it to Aaron.

Hopefully it will be on his website soon.

7. ### sle

didn't read the first message, but the whole deal is to:
a) determine CTD bond
b) get the current price of the CTD bond
c) calculate fwd price of the CTD bond using appropriate repo
d) calculate fwd yield of the CTD using the fwd price

if you do not have the current price and the repo, your answer will be significantly off. Alternatively, you can just calculate the yield of the theoretical deliverable bond (6% cpn etc).

8. ### futures_shark

The final goal that I am trying to achieve is to relate the \$TNX quote with the current futures contract price. Since those are the only live quotes I have available. If it's not possible with only these data I'll have to use an approximation

\$TNX is the cash yield on the most recently issued note, not the CTD.

based on the yield quoted by \$TNX I can calculate the CTD by pricing all bonds available for delivery. (using the original formulas)

However I believe this equation is definitely only an approximation. The spreadsheet from ramuk shows a price of exactly 100 at 6% yield, which is as expected. However this formula comes up with 100.65. There must be additional terms necessary or a totally different formula

Maybe something to do with duration & convexity, although I need to do more research on those topics to even know if they have anything to do with it.

9. ### sle

If you are trying to calculate the "futures" yield, you do need to know the CTD bond to do it well. If you do know the CTD bond, you can approximate by calculating the fwd bond price (futures * conversion) and then calculate the yield based on that price and delivery date. The yield will be a touch higher because of the basis (the seller has advantage), these days not as much (say 5 bps as of friday on TY). CTD does not change much these days, you can calculate it once a day only.

overall, there is a million ways to skin this particular cat. If I am reading your last message correctly, you are trying to use life quotes (i.e OTR 2s, 5s, 10s, bonds) to monitor the prices of the futures?

10. ### futures_shark

A yield based on the futures price is what I'm looking for.

I'm using data from the CBOT. So I have futures prices and the 10yr cash yield quote from is actually published by CBOE. According to CBOE \$TNX quotes the cash yield of the most recently auctioned 10yr.

I also have a list of deliverable bonds from the CBOT.

The very first post in this thread shows the formula that I am using to calculate bond prices.

I use this formula to price all deliverable bonds based on the cash yield quoted by \$TNX, which closed at 4.073 on Friday. Then choosing the lowest price ass the CTD. Using this method I came up with the note issued 5/15/2003 yielding 3 5/8 as the CTD

Then using the cash flow of this bond I use the same formula to calculate the implied yield based on Fridays closing price 112 20/32 and the conversion factor provided by CBOT.

The result is 4.292% which I believe to be incorrect since the current cash yield is 4.073%

#10     May 30, 2005
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