Explanation:
Let and be the positions in Bond A and Bond B. To perfectly hedge the position, the total dollar duration and dollar convexity of the combined portfolio must be zero. Portfolio Dollar Duration = 2 * 10 = 20 Portfolio Dollar Convexity = 2 * 60 = 120
Setting up the system of linear equations for a perfect hedge:
$20 + 6 V_A + 4 V_B = 0 \implies 6 V_A + 4 V_B = -20 \implies 3 V_A + 2 V_B = -10120` + 50 V_A + 20 V_B = 0 \implies 50 V_A + 20 V_B = -120 \implies 5 V_A + 2 V_B = -12$ (Equation 2)
Subtract Equation 1 from Equation 2:
$2 V_A = -2 \implies V_A = -1$ million.
Substitute into Equation 1:
$3(-1) + 2 V_B = -10 \implies -3 + 2 V_B = -10 \implies 2 V_B = -7 \implies V_B = -3.5$ million.
Since both values are negative, the bank needs to short 1 million of Bond A and short 3.5 million of Bond B.
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Q.52 A bank has a position of USD 2 million with a duration of 10 and a convexity of 60. To hedge against its position, the bank uses two bonds:
How can the bank hedge against its position?
A
Take a long position of USD 2 million in bond A and a long position of USD 8 million in bond B.
B
Take a short position of 1 million in bond A and a short position of 3.5 million in bond B.
C
Take a long position of USD 3 million in bond A and a long position of USD 9 million in bond.
D
Take a short position of USD 3 million in bond A and a long position of USD 9 million in bond B.
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