Question

An investor holds a bond portfolio with principal value $10,000,000 whose price and modified duration are respectively 112 and 9.21. He wishes to be hedged against a rise in interest rates by selling futures contracts written on a bond. Suppose the price of the cheapest-to-deliver issue is 105.2. The nominal amount of the futures contract is $100,000. The conversion factor for the cheapest-to-deliver is equal to 0.981. The cheapest-to-deliver has a modified duration equal to 8. Additionally, assume that if the yield of portfolio changes by 25 basis points, the yield of cheapest-to-deliver issue changes by 70 basis points. What is the number of futures contracts that the investor has to sell to hedge against a rise in interest rates?

Answer 1

Let N be the number f futures contracts that the investor has to sell to hedge against a rise in interest rates.

Let's assume that the yield of portfolio changes by 25 basis points. The change in value of the portfolio will be

= MD x Δi x V

= 9.21 x 0.25% x 112 x 10,000,000

= $ 25,788,000

Change in the value of the future contracts

= MD x Δi x V / CF

= 8 x 0.70% x 105.2 x 100,000 x N / 0.981

=  600,530.07N

Since we want this to be a perfect hedge, hence

The change in value of the portfolio = Change in the value of the future contracts

Hence,  25,788,000 = 600,530.07N

Hence, N = 42.94 = 43 (say, integral number) number of contracts.