My question is, how did people first come up with this formula?
$$CF = a (\frac{coupon}{2}+c+d)-b$$
Where $a,b,c,d$ are further defined as (strange looking) nonlinear functions of the bond's parameters.
How did they derive it? How do you go from these formula to the intuition of 6% in the end?
Computation method: https://www.cmegroup.com/trading/interest-rates/files/Calculating_U.S.Treasury_Futures_Conversion_Factors.pdf
Imagine wanting to create a deliverable bond future tradable on an exchange. We define which single bond is deliverable to it and use the forward price of the bond as the basis for the future. This works fine. I believe this is how Swedish government bond futures work.
Someone now suggests that there should be more than one deliverable bond into the futures contract. There is now a potential basket. An astute trader makes the point that some bonds will always have a higher price than others based on their coupon. If the future is just a forward price of one of those bonds it will always be of the bond with the lowest price (this is the cheapest to deliver); this makes no sense.
The suggestion is to standardise all the bonds in the basket to make them comparable. If the nominal coupon on the future is (arbitrarily) 6% then whatever price each bond has with a YTM of 6% acts as the conversion factor. If the futures price is 100 then multiplied by each bond's conversion factors this means Bond A might be deliverable at 100 * 1.01 = 101.0 for a futures price of 100, and Bond B might be deliverable at 100 * 0.99 = 99.0 for a futures price of 100. Bond B has a lower coupon (and lower price) relative to Bond A so this is a good standardisation. This is how UK government bond conversion factors are derived (except I believe the nominal coupon has been lowered to 3% to bring it more inline with market rates)
Now someone at CME goes, rather than use the YTM formula with its slight difficulties in handling fractional periods, lets just write out a standardised formula which gets very close to that. After all, it doesn't matter exactly what the conversion factors are (since the nominal coupon % is arbitrarily chosen anyway), just that there are some.
