Essentially the steps are:
1) Pick any bond which is eligible for delivery. (You should pick the Cheapest-to-Deliver, but Bloomberg's pay service is one of the few places that has that data. You can compute it yourself, but be prepared for several hours of work. Fortunately, CBOT publishes conversion factors for arbitrary bonds and notes that should get you pretty close to par. So, just pick one)
2) Get a price quote for that bond (I used Schwab to get a quote for a specific treasury)
3) Compute the Modified Duration and the accrued interest. (Excel and some simple math will get you those two values)
4) DV01 = ((.01 * ModifiedDuration)*(PriceOfBond + Accrued Interest)*.01)
5) Futures DV01 =( (Contract $ value)/1000)*DV01/Conversion Factor
That will give you the price move for a 1 basis point move in yield on the futures contract.
Do the same for the 10 year note, and then divide one by the other.
A typical DV01 for a 10 year note at 4.75% yield to maturity is about $80. A 2 year note is around $18.84. The two year contract is $2000 per point, whereas the 10 year is $1000 per point.
Taking into account the conversion factors for the specific bonds I chose, you end up with a 10 year Futures DV01 of $86.36, 2 year is $38.42. Or, about 2.25 2 years for every 1 10 year.
Long-winded enough? After all of that work, it's easier to just look at where CBOT will give you a price break. Unless you're doing HUGE trades, slight variances in your DV01 neutrality probably won't bother you much.
