Bond futures are contracts that allow investors to buy or sell a standardized amount of bonds at a specified price and date in the future. They are used to hedge interest rate risk, speculate on market movements, or arbitrage price differences.
However, not all bonds are equally suitable for delivery into a bond futures contract. The contract specifies certain criteria that the bonds must meet, such as maturity range, coupon rate, and issuer. These are called deliverable bonds. For example, the 10-year Treasury note futures contract allows the delivery of any U.S. Treasury note with at least 6.5 years and not more than 10 years to maturity as of the first day of the delivery month.
But even among the deliverable bonds, there are differences in coupon and maturity that affect their value and yield. To account for these differences, each deliverable bond is assigned a conversion factor that reflects its value relative to a hypothetical bond with a 6% coupon and the same maturity as the futures contract. The conversion factor is calculated as the price of the bond assuming a 6% yield to maturity, divided by 100.
The conversion factor is used to determine the delivery price of the bond, which is the amount that the seller of the futures contract receives from the buyer. The delivery price is equal to the product of the futures settlement price and the conversion factor, plus any accrued interest.
The conversion factor also helps the seller of the futures contract to choose the cheapest-to-deliver (CTD) bond, which is the bond that minimizes the cost of fulfilling the contract obligation. The CTD bond is usually the one with the lowest coupon and the longest maturity among the deliverable bonds, because it has the lowest price and the highest conversion factor. However, the CTD bond may change over time due to market fluctuations and changes in interest rates.
Basic Theory:
In the bond futures market, contracts are standardized, representing an agreement to buy or sell a bond at a future date. The deliverable bond is the specific bond that the seller must deliver to the buyer upon the contract’s expiration. Conversion factors come into play when considering that the deliverable bond might have a different face value than the standard contract.
The conversion factor is a measure of the bond’s relative value compared to the standard contract. It reflects the present value of the bond’s future cash flows, accounting for differences in coupon rates and maturity. A higher conversion factor indicates a lower present value and vice versa.
Procedures:
- Identify Deliverable Bonds: Determine the eligible deliverable bonds for a specific bond futures contract.
- Calculate Conversion Factors: Compute the conversion factor for each deliverable bond by considering its coupon rate, yield, and time to maturity.
- Calculate the Cheapest-to-Deliver (CTD): Identify the deliverable bond with the highest conversion factor as it is the cheapest for the short position to deliver.
- Calculate Duration: Use the Macaulay duration formula to calculate the duration of the bond portfolio, considering the weights of each deliverable bond based on its notional value and duration.
Comprehensive Explanation:
Let’s consider a scenario with three deliverable bonds (A, B, and C) for a bond futures contract. The relevant details are as follows:
- Bond A: Face Value = $1,000, Coupon Rate = 5%, Yield = 4%, Time to Maturity = 5 years
- Bond B: Face Value = $1,000, Coupon Rate = 6%, Yield = 5%, Time to Maturity = 7 years
- Bond C: Face Value = $1,000, Coupon Rate = 4%, Yield = 3%, Time to Maturity = 4 years
Calculations in Excel:
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- Calculate Conversion Factors:
=1/((1+Yield/2)^(2*Time to Maturity))
Apply this formula for each bond to get their respective conversion factors.
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- Identify Cheapest-to-Deliver (CTD):
The bond with the highest conversion factor is the CTD.
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- Calculate Duration:
=SUMPRODUCT(Notional Value * Macaulay Duration)
The Macaulay duration for each bond is calculated as:
=((Face Value * (1 + Yield/2)^(2 * Time to Maturity))/(2 * Yield) - Time to Maturity)/((1 + Yield/2)^(2 * Time to Maturity))
Apply these formulas for each bond and calculate the overall duration.
Results:
Assuming Notional Values are $500,000 for Bond A, $300,000 for Bond B, and $200,000 for Bond C:
- CTD: Bond B (highest conversion factor)
- Portfolio Duration: Calculated using the formulas above
Other Approaches:
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- Effective Duration:
Instead of Macaulay duration, use the effective duration formula, which considers the bond’s price sensitivity to changes in yield.
=(-1/Price) * (dPrice/dYield)
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- Modify Duration:
Adjust the Macaulay duration for changes in yield by incorporating the modified duration formula.
=Macaulay Duration / (1 + Yield/2)
