The manager of a government bond mutual fund knows that in 3 months $50 million of bonds will mature and the proceeds will be reinvested in other government bonds. The fund manager decides to use a Treasury bond futures contract that matures in 3 months to hedge the cost of reinvesting the $50 million. Suppose that the futures contract price is $109,750 and the duration of the bond portfolio in 3 months will be 5.20 years. The cheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 7% per annum coupon bond. The yield on this bond is currently 6.3% per annum, and the duration will be 6.4 years at the maturity of the futures contract. What is the position that the mutual fund manager should take in the Treasury bond futures contract?

Question

The manager of a government bond mutual fund knows that in 3 months $50 million of bonds will mature and the proceeds will be reinvested in other government bonds.  The fund manager decides to use a Treasury bond futures contract that matures in 3 months to hedge the cost of reinvesting the $50 million. Suppose that the futures contract price is $109,750 and the duration of the bond portfolio in 3 months will be 5.20 years. The cheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 7% per annum coupon bond. The yield on this bond is currently 6.3% per annum, and the duration will be 6.4 years at the maturity of the futures contract. What is the position that the mutual fund manager should take in the Treasury bond futures contract?

BlackTom AI · Accepted Answer

Question restatement: The fund manager will have $50 million maturing in 3 months and reinvested in government bonds. To hedge the reinvestment cost, they consider a T-bond futures contract maturing in 3 months, with futures price 109,750. The portfolio’s duration in 3 months is 5.20 years. The cheapest-to-deliver bond is a 20-year 7% coupon bond, with a yield of 6.3% and a duration of 6.4 years at the futures’ maturity. Determine the appropriate position in the Treasury bond futures.

Option analysis:
Option presented: Long 370 contracts.
- Step 1: Identify the hedging formula. For an approximate immunized hedge using duration-matched futures, the hedge ratio is h ≈ (D_p × V_p) / (D_f × V_f), where D_p is the portfolio duration, V_p is the value of the portfolio, D_f is the futures’ duration, and V_f is the value of one futures contract.
- Step 2: Input values. V_p is the value to be reinvested, which is $50,000,000. D_p is 5.20 years. D_f is 6.4 years (the duration of the CTD bond at futures maturity). V_f is the value per futures contract, which is the notional value of one contract. With a futures price of 109,750 and a standard par of $100,000 for a T-bond futures contract, V_f = $109,750 per contract.
- Step 3: Compute the denominator. V_f × D_f = 109,750 × 6.4 = 702,400.
- Step 4: Compute the numerator. V_p × D_p = 50,000,000 × 5.20 = 260,000,000.
- Step 5: Compute the hedge ratio. h ≈ 260,000,000 / 702,400 ≈ 370.32 contracts.

- Step 6: Rounding and direction. Since the fund will need to purchase bonds in 3 months, price increases of the cash bonds raise the cost of reinvestment. A long futures position increases in value when the futures price rises, offsetting higher cash-bond costs. Therefore, a long hedge is appropriate. Rounding to the nearest whole contract gives 370 contracts.

- Step 7: Conclusion about the option. The computed hedge requires approximately 370 futures contracts, and the position should be long to hedge the reinvestment cost.

Why the other potential interpretations would be incorrect (conceptual contrasts):
- A short futures position would hedge against falling bond prices, not against the risk of higher reinvestment costs, and would exacerbate losses if cash-bond prices rise.
- Using a different hedge ratio without matching D_p, D_f, and contract value could leave residual risk due to duration and price discrepancies; the duration-based approach aligns the sensitivity of the futures to the cash bond portfolio.
- If one incorrectly uses a cash-flow hedge or ignores the futures’ duration, the hedge would misalign the sensitivities and fail to offset the reinvestment cost risk.

In summary, the estimated hedge requires taking a long position in approximately 370 Treasury bond futures contracts to offset the duration and price risk of reinvesting the $50 million in 3 months.

BU.232.710.W1.SP25 Quiz #2

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