Duration Matching with a Swap

Consider a bank with the following balance sheet:

Value of Assets: V_A = $10,000

Duration of Assets: D_A = 4.2 yrs

Value of Liabilities: V_L = $20,000

a) Suppose you want to reduce duration gap to zero by using a swap. In particular, you can swap a portion of the 4yr loan @5% liability with a 4yr loan @Libor liability. The duration of the 4yr loan @Libor liability is 2 years. Find the swap size that reduces duration gap to zero.

Assets Value Duration 5yr loan @8% $6,000 4 5yr loan 25% $4,000 4.5 Liabilities Value Duration 5yr loan @ Libor $ 12,000 1.5 4yr loan @5% $ 8,000 3.5 Assets Value Duration 5yr loan @8% $6,000 4 5yr loan 25% $4,000 4.5 Liabilities Value Duration 5yr loan @ Libor $ 12,000 1.5 4yr loan @5% $ 8,000 3.5