Question

1) You are the manager of a bond portfolio of $10 million face value of bonds worth $9,448,546. The portfolio has a duration of 8.33. You plan to liquidate the portfolio in six months and are concerned about an increase in interest rates that would produce a loss on the portfolio. You would like to convert your portfolio to synthetic cash. A T-bond futures contract with the appropriate expiration is priced at 72 3/32 with a face value of $100,000, and an implied duration of 8.43 years. Use a yield beta of 1.0. Hints: (i) target duration is zero, (ii) the bond portfolio is worth $9,448,546: that’s the B in equation 6.4, (iii) the futures price (the “f” in equation 6.4) is 72 3/32 percent of $100,000 = $72,094.

a) Should you buy or sell futures? How many contracts should you use?

b) In six months, the bond portfolio has fallen in value to $8,952,597. The futures price is 68 16/32. Determine the profit from the transaction.

2) What if your target duration were 5?

a) Should you buy or sell futures? How many contracts should you use?

b) In six months, the portfolio has fallen in value to $8,952,597. The futures price is 68 16/32. Determine the profit from the transaction.

Answer 1

1) a) target duration is zero. so we are reducing current duration of portfolio to zero. we need to sell futures contracts to reduce the duration.

No. of contracts to sell = [(MDUR_T - MDUR_B)/MDUR_f]*(B/f)*yield beta

MDUR_T = target duration; MDUR_B = bond portfolio duration; MDUR_f = futures duration; B = current bond portfolio value; f = futures value

No. of contracts to sell = [(0-8.33)/8.43]*(9,448,546/72,094)*1 = (-8.33/8.43)*131.0587011401781‬*1 = -0.9881376037959668‬*131.0587011401781‬*1 = -129.50 or 130 contracts.

no. of contracts is negative because we need to sell the contract.

b) profit from transaction = no. of contracts sold*(current price - original price) + (original portfolio value - current portfolio value)

Profit from transaction = -130*(68,500 - 72,094) + (9,448,546 - 8,952,597) = -130*-3,594 + 495,949 = 467,220 + 495,949 = $963,169

2) a) target duration is 5. so we are reducing current duration of portfolio to 5. we need to sell futures contracts to reduce the duration.

No. of contracts to sell = [(MDUR_T - MDUR_B)/MDUR_f]*(B/f)*yield beta

MDUR_T = target duration; MDUR_B = bond portfolio duration; MDUR_f = futures duration; B = current bond portfolio value; f = futures value

No. of contracts to sell = [(5 - 8.33)/8.43]*(9,448,546/72,094)*1 = (-3.33/8.43)*131.0587011401781‬*1 = -0.395017793594306‬‬*131.0587011401781‬*1 = -51.77 or 52 contracts.

no. of contracts is negative because we need to sell the contract.

b) profit from transaction = no. of contracts sold*(current price - original price) + (original portfolio value - current portfolio value)

Profit from transaction = -52*(68,500 - 72,094) + (9,448,546 - 8,952,597) = -52*-3,594 + 495,949 = 186,888 + 495,949 = $682,837