Intuition and reasoning behind conversion factor calculation for bond futures

Question

I am currently reading the chapter on bond futures from J.C. Hull. The author states the procedure for calculating conversion factor as

The conversion factor for a bond is set equal to the quoted price the bond would have per dollar of principal on the first day of the delivery month on the assumption that the interest rate for all maturities equals 6% per annum (with semiannual compounding). The bond maturity and the times to the coupon payment dates are rounded down to the nearest 3 months for the purposes of the calculation. The practice enables the exchange to produce comprehensive tables. If, after rounding, the bond lasts for an exact number of 6-month periods, the first coupon is assumed to be paid in 6 months. If, after rounding, the bond does not last for an exact number of 6-month periods (i.e., there are an extra 3 months), the first coupon is assumed to be paid after 3 months and accrued interest is subtracted.

I have $3$ related questions regarding this procedure:

1. Why is this procedure of rounding off to three months done ? What is the logic behind it ? If I am not wrong looking at the time left to maturity of bond at start of delivery month is say $2$ years and $4$ months. I know the coupon the first coupon payment will happen next two months ( I assume coupons are given every $6$ months and are regular).

2. first coupon is assumed to be paid after 3 months and accrued interest is subtracted.
   Why is the accrued interest subtracted from the discounted value calculated, to get the quoted current price ? I have attached the example where the coupon cash payments are discounted and then accrued interest is subtracted.
   
   If I am not wrong the price calculated in example $2$ by discounting is the quoted price only ?

3. Why is the constant discounting rate of $6$ per cent used ? Why isn't the yields from the zero yield curve used. Wouldn't that also the eliminate the cheapest to deliver and bring all bonds in the nasket to more equal footing ? Does this have anything to do with point $1$.

PS: I googled around but could not get answer to my questions. I found this question related Conversion factor for bonds, but could not understand much.

PPS: I am really sorry for my naive question. I am a complete beginner and am having hard time with this section of the book. And am stuck on it. I am just looking for some intuitive explanation if a concrete one is complex.

Answer 1

I haven't calculated a conversion factor in a long time but it would appear that the answer is given in the quotation "to allow the exchange to produce a table of values". If you perform this ad hoc rounding in steps 1 and 2 then you reduce any arbitrary bond to an instrument that be determined from a table.

With regards to the 6% that is the de facto futures contract specification and has been for many years. When you pay $x$ for a bond future you are willing and expecting to accept delivery of a 6% bond for the maturity as also specified on the contract. The secondary artifact that only certain bonds exist that may or may not have a 6% coupon and precisely the right maturity means there is a need to convert your expected futures price to one that is fair for both parties to settle on another bond of different specification.

In the end a simpler and more transparent conversion of settlement prices benefits the exchange and traders since one can perform analytics with knowledge of cash values and expected returns.