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