Question

If you are a bank with liabilities of duration 4 years, how will you choose the composition of your assets to eliminate the interest rate risk that you would be exposed to if you had a duration mismatch on your balance sheet? Assume your assets can only be 3 year or 25 year zero coupon bonds.

• A. I would purchase a portfolio consisting for 95% of 3 year and for 5% of 25 year bonds with a total value equal to the value of my liabilities.
• B. I would purchase a portfolio consisting for 50% of 3 year for 50% of 25 year bonds with a total value equal to the value of my liabilities.
• C. I would purchase a portfolio consisting for 90% of 3 year and for 10% of 25 year bonds with a total value equal to the value of my liabilities.
• D. I would purchase a portfolio consisting for 10% of 3 year and for 90% of 25 year bonds with a total value equal to the value of my liabilities.

Answer 1

The correct answer is (a).

You want your overall portfolio to have 0 duration (to be duration neutral). If you purchase a portfolio of assets that has total value equal to the value of your liabilities, then you need to match the duration of the asset portfolio with the duration of the liabilities. The duration of the liabilities is 4 years. The duration of the asset portfolio is given by the weighted average of the two assets that constitute it. If w is the weight on the 3 year assets then (1-w) is the weight on the 25 year assets and setting the asset duration equal to 4 means that: w*3+(1-w)*25 = 4. Solving for w gives w= 0.954, which corresponds most closely to answer (a).