Homework 5_Bank Management

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Kent State University *

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56068

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Finance

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Jan 9, 2024

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1 5 rd Set of Homework (Due Date: December 6, 2023) Ch22 9. The duration of a 20-year, 8 percent coupon Treasury bond selling at par is 10.292 years. The bond’s interest is paid semiannually, and the bond qualifies for delivery against the Treasury bond futures contract. a. What is the modified duration of this bond? The modified duration is 10.292/1.04 = 9.896 years. b. What is the impact on the Treasury bond price if market interest rates increase 50 basis points? $100,000 = -9.896 x 0.005 x $100,000 = -$4,948.08. c. If you sold a Treasury bond futures contract at 95 and interest rates rose 50 basis points, what would be the change in the value of your futures position? P=- 9.896(0.005)$95,000=- $4,700.67 d. If you purchased the bond at par and sold the futures contract, what would be the net value of your hedge after the increase in interest rates? -$4,948.08-$4,700.67=-$247.41 20. A mutual fund plans to purchase $500,000 of 30-year Treasury bonds in four months. These bonds have a duration of 12 years and are priced at 96-08 (32nds). The mutual fund is concerned about interest rates changing over the next four months and is considering a hedge with T-bond futures contracts that mature in six months. The T- bond futures contracts are selling for 98-24 (32nds) and have a duration of 8.5 years. a. If interest rate changes in the spot market exactly match those in the futures market, what type of futures position should the mutual fund create? The mutual fund needs to buy futures contracts, thus entering a contract to buy Treasury bonds at 98-24 in four months. The fund manager fears a fall in interest rates (meaning the T-bonds price will increase) and by buying a futures contract, the profit from a fall in rates will offset a loss in the spot market from having to pay more for the securities.

2 b. How many contracts should be used? 12*$481,250/8.5*$98,750= 6.88 contracts Adjusted upwards to 7 contracts c. If the implied rate on the deliverables bond in the futures market moves 12 percent more than the change in the discounted spot rate, how many futures contracts should be used to hedge the portfolio? 6.88/1.12 = 6.14 contracts. This may be adjusted downward to 6 contracts d. What causes futures contracts to have a different price sensitivity than the assets in the spot markets? One reason for the difference in price sensitivity is that the futures contracts and the cash assets are traded in different markets. 21. Consider the following balance sheet (in millions) for an FI: Assets Liabilities Duration = 10 years $950 Duration = 2 years $860 Equity $90 a. What is the FI's duration gap? The duration gap is 10 - (860/950)(2) = 8.19 years b. What is the FI's interest rate risk exposure? The FI is exposed to interest rate increases. The market value of equity will decrease if interest rates increase. c. How can the FI use futures and forward contracts to put on a macrohedge? The FI can hedge its interest rate risk by selling future or forward contracts. d. What is the impact on the FI's equity value if the relative change in interest rates is an increase of 1 percent? That is, D R/(1+R) = 0.01. - 8.19(950,000)(0.01) = -$77,800

3 e. Suppose that the FI in part (c) macrohedges using Treasury bond futures that are currently priced at 96. What is the impact on the FI's futures position if the relative change in all interest rates is an increase of 1 percent? That is, D R/(1+R) = 0.01. Assume that the deliverable Treasury bond has a duration of nine years. -9(96,000)(0.01) = -$8,640 per futures contract. Since the macro hedge is a short hedge, this will be a profit of $8,640 per contract. f. If the FI wants to macrohedge, how many Treasury bond futures contracts does it need? To macro hedge, the Treasury bond futures position should yield a profit equal to the loss in equity value (for any given increase in interest rates). Thus, the number of futures contracts must be sufficient to offset the $77,800 loss in equity value. This will necessitate the sale of $77,800/8,640 = 9.005 contracts. Rounding down, to construct a macro hedge requires the FI to sell 9 Treasury bond futures contracts. 22. Refer again to problem 21. How does consideration of basis risk change your answers to problem 21? In problem 21, we assumed that basis risk did not exist. That allowed us to assert that the percentage change in interest rates ( D R/(1+R)) would be the same for both the futures and the underlying cash positions. If there is basis risk, then ( D R/(1+R)) is not necessarily equal to ( D R f /(1+R f )). If the FI wants to fully hedge its interest rate risk exposure in an environment with basis risk, the required number of futures contracts must reflect the disparity in volatilities between the futures and cash markets. a. Compute the number of futures contracts required to construct a macrohedge if [ D R f /(1+R f ) / D R/(1+R)] = br = 0.90 -8.19(950,000)/ (9)(96,000)(.90)= -10 contracts b. Explain what is meant by br = 0.90. br = 0.90 means that the implied rate on the deliverable bond in the futures market moves by 0.9 percent for every 1 percent change in discounted spot rates ( D R/(1+R)). c. If br = 0.90, what information does this provide on the number of futures contracts needed to construct a macrohedge? If br = 0.9 then the percentage change in cash market rates exceeds the percentage change in futures market rates. Since futures prices are less sensitive to interest rate shocks than cash prices, the FI must use more futures contracts to generate sufficient cash flows to offset the cash flows on its balance sheet position.

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