What is the $convexity for a long 100M 10Y and short barbell position?

What is the PnL for a rotational move of the yield curve based on the duration and convexity of each leg?

For the three bonds above, how can you combine 5Y and 20Y (barbell) to have the same $dur as 10Y? Maturity Yld dur $ amount $dur 5 1.50% -4.93 10 1.75% -9.83 100000000.00 $ 20 2.25% -19.56 (982,800,982.80) weight in 5 is W solving equation 100000000*w*(-4.93)+100000000*(1-w)*(-19.56)=-9828082.80 w*4.93+(1-w)*19.56=9.83 19.56-w*(19.56-4.93)=9.83 w=9.83/14.63 0.67 Summary: three-way butterfly postion, barbell of 5 and 20Y vs 10Y (short bullet) zero net dollar duration Twisting movement / rotation movement + Maturity Yld_before Yld_after Yld Change $ position $dur $con 5 1.50% 1.25% -0.25% $ (67,190,704.03) ####### 10 1.75% 1.50% -0.25% $ 100,000,000.00 20 2.25% 2.00% -0.25% $ (32,809,295.967) For the three bonds above, how can you combine 5Y and 20Y (barbell) to have the same $dur as 10Y? Maturity Yld dur $ amount $dur 5 1.50% -4.93 10 1.75% -9.83 100000000.00 $ 20 2.25% -19.56 (982,800,982.80) weight in 5 is W solving equation 100000000*w*(-4.93)+100000000*(1-w)*(-19.56)=-9828082.80 w*4.93+(1-w)*19.56=9.83 19.56-w*(19.56-4.93)=9.83 w=9.83/14.63 0.67 Summary: three-way butterfly postion, barbell of 5 and 20Y vs 10Y (short bullet) zero net dollar duration Twisting movement / rotation movement + Maturity Yld_before Yld_after Yld Change $ position $dur $con 5 1.50% 1.25% -0.25% $ (67,190,704.03) ####### 10 1.75% 1.50% -0.25% $ 100,000,000.00 20 2.25% 2.00% -0.25% $ (32,809,295.967)