Suppose one has a long position of F20 face value in 20-year Treasury bonds and wants to hedge this with short positions in both 10- and 30- year Treasury bonds. The prices and durations of 10-, 20-,...

Suppose one has a long position of F20 face value in 20-year Treasury bonds and wants to hedge this with short positions in both 10- and 30- year Treasury bonds. The prices and durations of 10-, 20-, and 30-year Treasury bonds are P10, DUR10, P20, DUR20, P30, and DUR30 and are assumed to be known. A regression of changes in the 20-year yield on changes in the 10- and 30-year yields is Δy20 = β0 + β1Δy10 + β2Δy30. The p-value of β0 is large and it is assumed that β0 is close enough to zero to be ignored. What face amounts F10 and F30 of 10- and 30-year Treasury bonds should be shorted to hedge the long position in 20-year Treasury bonds? (Express F10 and F30 in terms of the known quantities P10, P20, P30, DUR10, DUR20, DUR30, β1, β2, and F20.)