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Subject 5. Pricing Fixed-Income Forward and Futures Contracts PDF Download
Forward contracts on fixed-income securities are priced and valued in a virtually identical manner to their equity counterparts. To price a fixed-income forward contract, take the bond price, subtract the present value of the coupons over the life of the contract, and compound this amount at the risk-free rate to the expiration date of the contract. If the bond price is a clean price, add the accrued interest AI.

F0(T) = FV0,T[B0(T+Y) + AI0 - PVCI0,T]

where:

  • B0(T+Y) is the clean price at Time 0 of a bond that matures at Time T + Y.
  • AI0 is the accrued interest at Time 0.
  • PVCI0,T is the present value of all coupon payments from Time 0 to Time T.

In some fixed-income futures contracts, more than one bond can be delivered by the seller. A conversion factor (CF(T)) is often used to make all deliverable bonds roughly equal in price. The quoted futures price is:

QF0(T) = [1/CF(T)] {FV0,T[B0(T + Y) + AI0] - AIT - FVCI0, T}

The value of a fixed income forward contract is the present value of difference in forward prices.

Vt(T) = PVt, T[Ft(T) - F0(T)]

Example

Consider a bond with semi-annual coupons.

  • The par value is $1,000.
  • The coupon rate is 6%. This implies that each coupon is $30.
  • It has a current maturity of 593 days.
  • It pays four coupons: the next occurs in 47 days, followed by one in 229 days, 411 days, and 593 days.
  • The bond price is $994.45. This includes accrued interest.
  • The risk-free interest rate is 4.75%.
  • The forward contract expires in 305 days.

Analysis:

  • T = 305
  • Y = 288, which means the bond has 288 days remaining after the forward contract expires.
  • Only the first two coupons occur during the life of the forward contract. The present value of the coupons is: $30/(1.0475)47/365 + $30/(1.0475)229/365 = $58.96.
  • The forward price if the contract is initiated now is F(0, T) = ($994.45 - $58.96)(1.0475)305/365 = $972.48. This means that we shall be able to enter into this contract to buy the bond in 310 days at the price of $972.48.
  • Suppose it is 30 days later and the new bond price is $985.14. The risk-free rate is now 5.75%.

    • The present value of the remaining coupons is $30/(1.0575)17/365 + $30/(1.0575)199/365 = $59.02.
    • The value of the forward contract is thus $985.14 - $59.02 - $972.48/(1.0575)275/365 = -$6.25.
    • The contract has gone from a zero value at the start to a negative value, primarily because of the decrease in the price of the underlying bond.

User Contributed Comments 2

User Comment
BB465 In the previous section they use 360 days, but 365 here, which is correct?
vncld Bonds are usualy denoted on a 360 days basis
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I used your notes and passed ... highly recommended!
Lauren

Lauren

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