Duration hedge ratio for interest rate futures

CFA Level III Derivatives
#1

The formula for the duration hedge ratio for interest rate futures to protect against interest rate risk is: H = (PP* DP)/ (PF * DF) Where: PP is the forward value the fixed-income portfolio being hedged (at the maturity date of the hedge) DP is the duration of the portfolio at the maturity date of the hedge PF is the futures contract price DF is the duration of the asset underlying the futures contract at the maturity date of the contract

My question is why is the duration of the asset underlying the futures contract used in DF rather than the duration of the underlying asset plus the term to maturity for the futures contract?

Thank you very much

#2

you need to know how many futures contracts to buy / sell to get to your target duration.

for that only the duration of futures means anything

#3

Sorry I don’t really understand your answer… My question was why is the duration of the asset (bond) underlying the futures contract used rather than the the duration of the asset (bond) underlying the futures contract plus the term to maturity of the futures

#4

you are using the futures contract for the purpose of hedging. if you owned / bought / sold the underlying behind the futures contract - that affects the numerator (your portfolio). Your hedge (denominator) is only the futures contract.

#5

I still don’t think you understand my question/formula provided.

The numerator is for a bond portfolio (lets say a $2 million porfolio of bonds that I currently own)

The denominator is for an interest rate futures hedging tool (lets say by X contracts of $100,000 bond futures, where the futures contract is due for delivery in 8 months)

My question is -why in the denominator for the forumula is the duration of the $100,000 underlying bond only rather than the duration of the $100,000 underlying bond plus the 8 months?

NB

See link here

https://au.mathworks.com/help/fininst/examples/managing-interest-rate-risk-with-bond-futures.html

It states " Since bond futures derive their value from the underlying instrument, the duration of a bond futures contract is related to the duration of the underlying bond. ". Not sure why this is though…

#6

The answer will be that duration of the underlying is different than duration of futures contract. Does that answer your q?

#7

Yes they are different - but logically/theoretically why?