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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4 31. An FI has assets denominated in British pounds of $125 million and pound liabilities of $100 million. The exchange rate of dollars for pounds is currently $1.60/£. a. What is the FI's net exposure? The next exposure is $125 million - $100 million = $25 million b. Is the FI exposed to a dollar appreciation or depreciation? The FI is exposed to dollar appreciation, or declines in the pound relative to the dollar. c. How can the FI use futures or forward contracts to hedge its FX rate risk? The FI can hedge its FX rate risk by selling forward or futures contracts in pounds, assuming the contracts are quoted as $/£, that is in direct quote terms in the U.S. d. If a futures contract is currently trading at $1.55/£, what is the number of futures contracts that must be utilized to fully hedge the FI's currency risk exposure? Assume the contract size on the British pound futures contract is £62,500. Nf= ($25million/1.55)/ £62,500 = 258.0645 = 258 pound sterling futures contracts e. If the British pound exchange rate falls from $1.60/£ to $1.50/£, what will be the impact on the FI's cash position? The cash position will experience a loss if the pound depreciates in terms of the U.S. dollar. The loss would be equal to (($25 million/$1.60) x 1.50) - $25million = $23,437,500 - $25 million = -$1,562,500 f. If the British pound exchange rate falls from $1.55/£ to $1.45/£, what will be the impact on the FI's futures position? £62,500 x D ƒ I = -258(£62,500)(-$1.45-$1.55) = $1,612,500 g. Using the information in parts (e) and (f ), what can you conclude about basis risk? In cases where basis risk occurs, a perfect hedge is not possible. 32. An FI is planning to hedge its one-year, 100 million Swiss franc (SF)-denominated loan against exchange rate risk. The current spot rate is $0.60/SF. A 1-year SF futures

5 contract is currently trading at $0.58/SF. SF futures are sold in standardized units of SF125,000. a. Should the FI be worried about the SF appreciating or depreciating? The FI should be worried about the SF depreciation because it will provide fewer dollars per SF. b. Should it buy or sell futures to hedge against exchange rate risk exposure? The FI should sell the SF futures contracts to hedge this exposure. c. How many futures contracts should it buy or sell if a regression of past spot exchange rates on changes in future exchange rates generates an estimated slope of 1.4? = (100 million * 1.40)/ 125,000 = 1120 contract d. Show exactly how the FI is hedged if it repatriates its principal of SF100 million at year-end, the spot exchange rate of SF at year-end is $0.55/SF, and the forward exchange rate is $0.5443/SF. Gain from future contract = (.55 - .5443) * 125000*1120 = 798,000 Less value $ =$100 million * .6 = $60 million Less value at year end = $100 million * .55 = $55 million Decreases value asset = 60 – 55 = 5 million (.58 - .5443) * 125,000 *1,120 = 4,998,000 5,000,000 – 4,998,000 = 2,000 maximum net loss by hedging 37. What is the gain on the purchase of a $20 million credit forward contract with a modified duration of seven years if the credit spread between a benchmark Treasury bond and a borrowing firm’s debt decreases by 50 basis points? =-7 (-.0050) *20,000,000= $700,000