Pricing and Valuation of Fixed Income Forwards

Derivatives 4 min readModule 3 of 7Save

What you'll learn

LOS 28.d
Describe how fixed income forwards and futures are priced, and calculate and interpret their no arbitrage value.

Forwards and Futures on Fixed Income Securities

In order to calculate the no arbitrage forward price on a coupon paying bond, we can use the same formula as we used for a dividend paying stock or portfolio, simply substituting the present value of the expected coupon payments (PVC) over the life of the contract for PVD, or the future value of the coupon payments (FVC) for FVD, to get the following formulas:

F P_{fixed income} = (S_{0} - P V C) (1 + R_{f})^{T} = S_{0} (1 + R_{f})^{T} - F V C

The value of the forward contract prior to expiration is as follows:

V_{t} (long) = (S_{t} - P V C_{t}) - \frac{F P}{( 1 + R _{f} ) ^{T - t}}

If given the current forward price ( $F P_{t}$ ) on the same underlying and with the same maturity:

V_{t} (long) = \frac{F P _{t} - F P}{( 1 + R _{f} ) ^{T - t}}

In our examples, we assume that the spot price of the underlying coupon paying bond includes accrued interest. For fixed income contracts, use a 365 day basis to calculate T if the contract maturity is given in days.

Worked example: Calculating the price of a forward on a fixed income security

Calculate the price of a 250 day forward contract on a 7% U.S. Treasury bond with a spot price of $1,050 (including accrued interest) that has just paid a coupon and will make another coupon payment in 182 days. The annual risk free rate is 6%.

Solution

Remember that U.S. Treasury bonds make semiannual coupon payments, so:

C P V C = \frac{$1 , 000 ( 0.07 )}{2} = $35.00 = \frac{$35.00}{1.0 6 ^{182/365}} = $34.00

The forward price of the contract is therefore:

F P = ($1, 050 - $34.00) 1.0 6^{250/365} = $1, 057.37

Worked example: Calculating the value of a forward on a fixed income security

After 100 days, the value of the bond in the previous example is $1,090. Calculate the value of the forward contract on the bond to the long position, assuming the risk free rate is 6.0%.

Solution

There is only one coupon remaining (in 82 days) before the contract matures (in 150 days), so:

P V C V_{100} (long) = \frac{$35.00}{1.0 6 ^{82/365}} = $34.54 = $1, 090 - $34.54 - \frac{$1 , 057.37}{1.0 6 ^{150/365}} = $23.11

If given the forward price directly, the value (to the long party) is simply the present value of the difference between the current forward price and the original, locked in forward price.

Worked example: Value of a fixed income forward contract after inception

Louise Michelle entered into a 250 day forward contract on a 7% U.S. Treasury bond at a forward price of $1057.37. 100 days later, the forward price has changed to $1081.04. The risk free rate is 6%.

Calculate the value to Michelle of the forward contract.

Solution

value = \frac{1081.04 - 1057.37}{1.0 6 ^{150/365}} = $23.11

Bond futures contracts often allow the short an option to deliver any of several bonds, which will satisfy the delivery terms of the contract. This is called a delivery option and is valuable to the short. Each bond is given a conversion factor that is used to adjust the long’s payment at delivery so the more valuable bonds receive a larger payment.

These factors are multipliers for the futures price at settlement. The long pays the futures price at expiration, multiplied by the conversion factor (CF).

Bond prices in some countries are quoted as clean prices. At settlement, the buyer actually pays the clean price plus an accrued interest, or the full price.

accrued interest full price = \frac{days since last coupon payment}{days between coupon payments} \times coupon amount = clean price + accrued interest = clean price + A I_{0}

The futures price can then be calculated as:

F P = (full price) (1 + R_{f})^{T} - F V C - A I_{T}

$A I_{T}$ is the accrued interest at maturity of the futures contract.

The quoted futures price adjusts this price based on the CF as follows:

QF P = \frac{F P}{C F} = [(full price) (1 + R_{f})^{T} - F V C - A I_{T}] \frac{1}{C F}

Worked example: Calculating the price of a Treasury bond futures contract

Suppose that you need to calculate the quoted futures price of a 1.2 year Treasury bond futures contract. The cheapest to deliver bond is a 7% semi annual coupon T bond with exactly 10 years remaining to maturity and a quoted price of $1,040 with a conversion factor of 1.13. There is currently no accrued interest because the bond has just paid a coupon.

The annual risk free rate is 5%. The accrued interest on the bond at maturity of the futures contract will be $14.

Solution

The full price of the bond = $1,040 quoted price + $0 accrued interest = $1,040. The semiannual coupon on a single, $1,000 face value 7% bond is $35. A bondholder will receive one payment 0.5 years from now (when there are 0.7 years left to maturity of the futures contract) and one payment 1 year from now (when there are 0.2 years until maturity).

The future value of these coupons at the end of 1.2 years (the expiration date) is:

F V C = ($35 \times 1.0 5^{0.7}) + ($35 \times 1.0 5^{0.2}) = $71.56

The quoted futures price is then:

QF P = [($1, 040 \times 1.0 5^{1.2}) - $71.56 - $14] \frac{1}{1.13} = $900.13

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