The average duration of the loans is 10 years. The average duration of the deposits is 3 years. What is the number of T-bond futures contracts necessary to hedge the balance sheet if the duration of the deliverable bonds is 9 years and the current price of the futures contract is $96 per $100 face value and if basis risk shows that for every 1 percent shock to interest rates, i.e., ΔR/(1 + R) = 0.01, the implied rate on the deliverable bonds in the futures market increases by 1.1 percent, i.e., ΔR_f/(1 + R_f) = 0.011?

Question

Quizplus · Accepted Answer

First, we need to calculate the duration of the balance sheet:
Duration of loans = 10 years
Duration of deposits = -3 years (negative because they are a liability)
Duration of balance sheet = (Duration of loans - Duration of deposits) = 13 years
Next, we need to calculate the ratio of the notional value of the balance sheet to the notional value of T-bond futures contracts:
Ratio = Notional value of balance sheet / (Futures price x $100 face value x Conversion factor)
The conversion factor is the duration of the T-bond futures contract divided by the duration of the deliverable bonds, which in this case is 9 years / 13 years = 0.6923.
Using the current futures price of $96 per $100 face value, we get:
Ratio = ($13,000,000 loans - $3,000,000 deposits) / ($96 x $100 x 0.6923) = 1,888 contracts
Finally, we need to adjust for basis risk. A 1% shock to interest rates corresponds to a change in the futures implied rate of 0.011 x 1.1 = 0.0121. Therefore, we need to buy additional futures contracts to offset the potential loss from a 1% increase in interest rates. The additional contracts needed are:
Additional contracts = Ratio x Duration x Basis risk = 1,888 x 13 x 0.0121 = 308
Total contracts needed = Ratio + Additional contracts = 1,888 + 308 = 2,196, which is closest to choice B: 1,888 contracts.

The Average Duration of the Loans Is 10 Years