You have a long position in $100M 30-Year Bond, How do you limit your exposure to $50M using future contracts

Answer

Limit exp= Hedge Away $50= Future Contract goes opposite direction when rate moves=short some future contracts.

Bond Price =Sum of Z(0,i), Price Move= 1st Derivative= n*wi (wights) so wieghted Average of "Duration"=D

Two Future Contract very liquid on CBOT: Bond Future and Euro Dollar future.
CTD= Cheapest to Deliver Bond Notion $100,000 each and EuroDollar short term 3-month 

50M/100K=500 Contracts * D/Dc, Dc=duration of CTD.