READing 39 : INTEREST RATE FUTURES

Day Count Conventions And Quotations

        Day count conventions play a role when computing the interest that accrues on a fixed income security. When a bond is purchased, the buyer must pay any accrued interest earned through the settlement date. The most common day count conventions are Actual/Actual, 30/360, and Actual/360.

Quotations for T-Bills

        T-bills are quoted on a discount rate basis. A T-bill with a $100 face value with n days to maturity and a cash price of Y is quoted as:

Quotations for T-Bonds

        For a U.S. Treasury bond, the dirty price is the price that the seller of the bond must be paid to give up ownership. It includes the present value of the bond plus the accrued interest. Conversely, the clean price is the dirty price less accrued interest.

Treasury Bond Futures

        Since deliverable bonds have very different market values, the Chicago Board of Trade (CBOT) has created conversion factors. Conversion factors are supplied by the CBOT on a daily basis. They are calculated as:
                 (bond discounted price − accrued interest) / face value
        The conversion factor system is not perfect and often results in one bond that is the cheapest (or most profitable) to deliver. The cheapest-to-deliver (CTD) bond is the bond that minimizes the following:
                 quoted bond price − (quoted futures price × conversion factor)
        When the yield curve is not flat, there is a single bond that is the cheapest-to-deliver (CTD). When the yield curve is upward sloping, CTD bonds tend to have longer maturities. When the yield curve is downward sloping, CTD bonds tend to have shorter maturities.

Treasury Bond Futures Price

        The theoretical price for a T-bond futures contract is calculated as:
                 (cash futures price − accrued interest) / conversion factor

Eurodollar Futures

        Eurodollar contracts are based on LIBOR and are quoted on a discount rate basis. If Z is the quoted price for a euro-dollar futures contract, the contract price is:
                 euro-dollar futures price = $10,000 × [100 − (0.25) × (100 − Z)]
        Long-dated euro-dollar contracts must be adjusted for convexity before being used to estimate the corresponding forward rates. As the maturity of the futures contract increases, the necessary convexity adjustment increases.

LIBOR zero curve

        Forward rates implied by convexity-adjusted euro-dollar futures can be used to produce a LIBOR spot curve. The following equation is used to generate the shape of the futures rate curve:

         where:
         R_i = spot rate corresponding with T_i periods
         R_Forward = the forward rate between T₁ and T₂

Duration-Based Hedging

        Duration can be used to compute the number of futures contracts needed to implement a duration-based hedging strategy. The duration-based hedge ratio can be expressed as follows:

Limitations of Duration

        The effectiveness of duration-based hedging strategies is limited when there are large changes in yield or nonparallel shifts in the yield curve.