 Why is bond future's hedge ratio calculated as HR = ΔP / ΔCTD * CF?

I don’t understand why hedge ratio HR = ΔP / ΔCTD * CF. My calculation shows it should be HR = ΔP / ΔCTD.

Let’s start from ΔP = (HR)(ΔF) where P is the value of your portfolio. This formula holds only if F is defined as the value of CTD bond, because one contract involves delivery of exact 1 unit of CTD bond. Therefore ΔP = (HR)(ΔF) = (HR)(ΔCTD) which gives HR = ΔP / ΔCTD.

The formula ΔP = (HR)(ΔF) doesn’t hold if F is defined as the future settlement price. This is because future settlement price is not the real amount of money transfered at delivery. It has to be adjusted by CF according to this equation “Principal invoice amount =(Futures settlement price/100) × CF × Contract size”

One example - Assume CTD bond is traded at \$80. The conversion factor (CF) is 0.8 and future settlement price is \$100. In this example the short side delivers 1 unit of CTD bond and receives \$100*0.8=\$80 at maturity date. Now you have 1 unit of CTD bond and want to use bond future to hedge your portfolio. Obviously the hedge ratio should be 1 (because you long 1 CTD so you need 1 contract to hedge it, 1 contract = 1 CTD). Using the formula provided by CFA material, HR = ΔP / ΔCTD * CF = ΔCTD / ΔCTD * CF = CF = 0.8, this doesn’t make sense.

I would appreciate if anyone could help double check the logic in this chapter. Thanks

No, it doesn’t.

One contract involves deliver of 1 unit if the theoretical underlying bond, not 1 unit of the CTD bond.

That’s the point of the conversion factor, CF.

Many thanks!

Now I am a little confused, because CFA mentioned this formula

Principal invoice amount = (Futures settlement price / 100) × CF × Contract size

Again in my example before, CTD bond is traded at \$80, CF is 0.8 and future settlement price is \$100. The formula above gives the principal invoice amount (assume contract size is \$100)

Principal invoice amount = (Futures settlement price / 100) × CF × Contract size = 100/100 * 0.8 * \$100 = \$80

So that means the long side pays short side \$80 on maturity, which is equivalent to only 1 unit of CTD bond. But you’re saying the short side should deliver 1/CF=1/0.8=1.25 unit of CTD bond, right? There seems to be a conflict, can you explain more on this?

Thanks!