# Financial-Institutions Management. Solutions The following are the foreign currency positions of an FI, expressed in the foreign currency.

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1 Solutions 4 Chapter 14: oreign Exchange Risk 8. The following are the foreign currency positions of an I, expressed in the foreign currency. Currency Assets Liabilities X Bought X Sold Swiss franc (S) S134,394 S53,758 S10,752 S16,127 British pound ( ) 30,488 13,415 9,146 12,195 Japanese yen ( ) 7,075,472 2,830,189 1,132,075 8,301,887 The exchange rate of dollars per Ss is.9301, of dollars per British pounds is , and of dollars per yen is The following are the foreign currency positions converted to dollars. Currency Assets Liabilities X Bought X Sold Swiss franc (S) \$125,000 \$50,000 \$10,000 \$15,000 British pound ( ) \$50,000 \$22,001 \$14,999 \$20,000 Japanese yen ( ) \$75,000 \$30,000 \$12,000 \$88,000 a. What is the I s net exposure in Swiss francs stated in Ss and in dollars? Net exposure in stated in Ss = S134,394 - S53,758 + S10,752 - S16,127 = S75,261 Net exposure in stated in \$s = \$125,000 - \$50,000 + \$10,000 - \$15,000 = \$70,000 1

2 b. What is the I s net exposure in British pounds stated in pounds and in dollars? Net exposure in British pounds = 30,488-13, ,146-12,195= 14,024 Net exposure in \$s = \$50,000 - \$22,001 + \$15,000 - \$20,000 = \$22,999 c. What is the I s net exposure in Japanese yen stated in yen and in dollars? Net exposure in Japanese yen = 7,075,472-2,830, ,132,075-8,301,887= - 2,924,529 Net exposure in \$s = \$75,000 - \$30,000 + \$12,000 - \$88,000 = -\$31,000 d. What is the expected loss or gain if the S exchange rate appreciates by 1 percent? State you answer in Swiss francs and dollars. If assets are greater than liabilities, then an appreciation of the foreign exchange rates will generate a gain = S75,261 x 0.01 = S or \$70,000 x 0.01 = \$ e. What is the expected loss or gain if the exchange rate appreciates by 1 percent? State you answer in pounds and dollars. Gain = 14,024 x 0.01 = \$140 or \$22,999 x 0.01 = \$230 f. What is the expected loss or gain if the exchange rate appreciates by 2 percent? State you answer in yen and dollars. Loss = - 2,924,529 x 0.02 = -\$58,491 or -\$31,000 x 0.02 = -\$620 2

3 10. City Bank issued \$200 million of one-year CDs in the United States at a rate of 6.50 percent. It invested part of this money, \$100 million, in the purchase of a one-year bond issued by a U.S. firm at an annual rate of 7 percent. The remaining \$100 million was invested in a one-year Brazilian government bond paying an annual interest rate of 8 percent. The exchange rate at the time of the transaction was Brazilian real 1/\$. a. What will be the net return on this \$200 million investment in bonds if the exchange rate between the Brazilian real and the U.S. dollar remains the same? Cost of funds = x \$200 million = \$13 million Return on U.S. loan = 0.07 x \$100 million = \$ 7,000,000 Return on Brazilian bond = (.08 x Real 100 m)/1.00 = \$ 8,000,000 Total interest earned = \$15,000,000 Net return on investment = \$15 million - \$13 million/\$200 million = 1.00 percent. b. What will be the net return on this \$200 million investment if the exchange rate changes to real 1.20/\$? Cost of funds = x \$200 million = \$13,000,000 Return on U.S. loan = 0.07 x \$100 million = \$ 7,000,000 Return on Brazilian bond = (0.08 x Real 100m)/1.20 = \$ 6,666,667 Total interest earned = \$13,666,667 Net return on investment = \$13,666,667 - \$13,000,000/\$200,000,000 = 0.33 percent. Consideration should be given to the fact that the Brazilian bond was for Real100 million. Thus, at maturity the bond will be paid back for Real100 million/1.20 = \$83,333, Therefore, the strengthening dollar will have caused a loss in capital (\$16,666,666.67) that far exceeds the interest earned on the Brazilian bond. 3

4 c. What will be the net return on this \$200 million investment if the exchange rate changes to real 0.80/\$? Cost of funds = x \$200 million = \$13,000,000 Return on U.S. loan = 0.07 x \$100 million = \$ 7,000,000 Return on Brazilian bond = (.08 x Real 100m)/0.80 = \$10,000,000 Total interest earned = \$17,000,000 Net return on investment = \$17,000,000 - \$13,000,000/\$200,000,000 = 2.00 percent. Consideration should be given to the fact that the Brazilian bond was for Real100 million. Thus, at maturity the bond will be paid back for Real100 million/0.80 = \$125,000,000. Therefore, the strengthening Real will have caused a gain in capital of \$25,000,000 in addition to the interest earned on the Brazilian bond. 11. Sun Bank USA purchased a 16 million one-year euro loan that pays 12 percent interest annually. The spot rate of U.S. dollars per euro is Sun Bank has funded this loan by accepting a British pound-denominated deposit for the equivalent amount and maturity at an annual rate of 10 percent. The current spot rate of U.S. dollars per British pound is a. What is the net interest income earned in dollars on this one-year transaction if the spot rates of U.S. dollars per euro and U.S. dollars per British pound at the end of the year are 1.50 and 1.85?. Loan amount = 16 million x 1.40 = \$22.4 million Deposit amount = \$22.4m/1.60 = 14,000,000 Interest income at the end of the year = 16m x 0.12 = 1.92m x 1.50 = \$2,880,000 Interest expense at the end of the year = 14,000,000 x 0.10 = 1,400,000 x 1.85 = \$2,590,000 Net interest income = \$2,880,000 - \$2,590,000 = \$290,000 4

5 b. What should be the spot rate of U.S. dollars per British pound at the end of the year in order for the bank to earn a net interest margin of 4 percent? A net interest margin of 4 percent would imply \$22,400,000 x 0.04 = \$896,000. The net cost of deposits should be \$2,880, ,000 = \$1,984,000. Pound rate = \$1,984,000/ 1,400,000 = \$1.4171/. Thus, the pound should be selling at \$1.4171/ in order for the bank to earn 4 percent. c. Does your answer to part (b) imply that the dollar should appreciate or depreciate against the pound? The dollar should depreciate against the pound. Each pound gives fewer dollars. d. What is the total effect on net interest income and principal of this transaction given the end-of-year spot rates in part (a)? Interest income and loan principal at year-end = ( 16m x 1.12) x 1.50 = \$26,880,000 Interest expense and deposit principal at year-end = ( 14m x 1.10) x 1.85 = \$28,490,000 Total income = \$26,880,000 - \$28,490,000 = -\$1,610, Bank USA just made a one-year \$10 million loan that pays 10 percent interest annually. The loan was funded with a Swiss franc-denominated one-year deposit at an annual rate of 8 percent. The current spot rate is S1.60/\$1. a. What will be the net interest income in dollars on the one-year loan if the spot rate at the end of the year is S1.58/\$1? Interest income at year-end = \$10m x 0.10 = \$1,000,000. 5

6 Interest expense at year-end = (S16,000,000 x 0.08)/1.58 = S1,280,000/1.58 = \$810, Net interest income = \$1,000,000 - \$810, = \$189, b. What will be the net interest return on assets? Net interest return on assets = \$189,873.42/\$10,000,000 = or 1.90 percent. c. How far can the S/\$ appreciate before the transaction will result in a loss for Bank USA? Exchange rate = S1,280,000/\$1,000,000 = S1.28/\$, appreciation of percent. d. What is the total effect on net interest income and principal of this transaction given the endof-year spot rates in part (a)? Interest income and loan principal at year-end = \$10m x 1.10 = \$11,000,000. Interest expense and deposit principal at year-end = (S16,000,000 x 1.08)/1.58 = S17,280,000/1.58 = \$10,936, Total income = \$11,000,000 - \$10,936, = \$63, What are the two primary methods of hedging X risk for an I? What two conditions are necessary to achieve a perfect hedge through on-balance-sheet hedging? What are the advantages and disadvantages of off-balance-sheet hedging in comparison to on-balance-sheet hedging? The manager of an I can hedge using on-balance-sheet techniques or off-balance-sheet techniques. Onbalance-sheet hedging requires matching currency positions and durations of assets and liabilities. If the duration of foreign-currency-denominated fixed rate assets is greater than similar currency 6

7 denominated fixed rate liabilities, the market value of the assets could decline more than the liabilities when market rates rise and therefore the hedge will not be perfect. Thus, in matching foreign currency assets and liabilities, not only do they have to be of the same currency, but also of the same duration in order to have a perfect hedge. Advantages of off-balance-sheet X hedging: The use of off-balance-sheet hedging devices, such as forward contracts, enables an I to reduce or eliminate its X risk exposure without forfeiting potentially lucrative transactions. On-balance-sheet transactions result in immediate cash flows, whereas off-balance-sheet transactions result in contingent future cash flows. Therefore, the up-front cost of hedging using off-balance-sheet instruments is lower than the cost of on-balance-sheet transactions. Moreover, since on-balance-sheet transactions are fully reflected in financial statements, there may be additional disclosure costs to hedging on the balance sheet. Off-balance-sheet hedging instruments have been developed for many types of risk exposures. or currency risk, forward contracts are available for the majority of currencies at a variety of delivery dates. Moreover, since the forward contract is negotiated over the counter, the counterparties have maximum flexibility to set terms and conditions. Disadvantages of off-balance-sheet X hedging: There is some credit risk associated with off-balance-sheet hedging instruments since there is some possibility that the counterparty will default on its obligations. This credit risk exposure is exacerbated in negotiated markets such as the forward market, but mitigated for exchange-traded hedging instruments such as futures contracts. 7

8 15. North Bank has been borrowing in the U.S. markets and lending abroad, thus incurring foreign exchange risk. In a recent transaction, it issued a one-year, \$2 million CD at 6 percent and funded a loan in euros at 8 percent. The spot rate for the euro was 1.45/\$1 at the time of the transaction. a. Information received immediately after the transaction closing indicated that the euro will change to 1.47/\$1 by year-end. If the information is correct, what will be the realized spread on the loan inclusive of principal? What should have been the bank interest rate on the loan to maintain the 2 percent spread? Amount of loan in = \$2 million x 1.45 = 2.9 million. Interest and principal at year-end = 2.9m x 1.08 = 3.132m/1.47 = \$2,130, Interest and principal of CDs = \$2m x 1.06 = \$2,120,000 Net interest income = \$2,130, \$2,120,000 = \$10, Net interest margin = \$10,612.24/2,000,000 = or 0.53 percent. In order to maintain a 2 percent spread, the interest and principal earned at 1.47/\$ should be: 2.9m.(1 + x)/1.47 = \$2.16m. (Because (\$2.16m. - \$2.12m.)/\$2.00m. = 0.02, or 2%). Therefore, (1 + x) = (\$2.16m. x 1.47)/ 2.9m. = , and x = or 9.49 percent, or the bank should have charged a rate of 9.49 percent on the loan. b. The bank had an opportunity to sell one-year forward euros at 1.46/\$. What would have been the spread on the loan if the bank had hedged forward its foreign exchange exposure? Net interest income if hedged = 2.9m. x 1.08 = 3.132m./1.46 = \$2.1452m. - \$2.12m. = \$ million, or \$25, Net interest margin = \$0.0252m./\$2m. = , or 1.26 percent c. What would have been an appropriate change in loan rates to maintain the 2 percent spread if the bank intended to hedge its exposure using the forward contracts? To maintain a 2 percent spread: 2.9m.(1 + x)/1.46 = \$2.16m. => x = 8.74 percent 8

9 The bank should increase the loan rate to 8.74 percent and hedge with the sale of forward s to maintain a 2 percent spread. 9

10 Chapter 14: Purchasing Power and Interest Rate Parity 19. Suppose that the current spot exchange rate of U.S. dollars for Australian dollars, S US\$/A\$, is.7590 (i.e., dollars, or 75.9 cents, can be received for 1 Australian dollar). The price of Australian-produced goods increases by 5 percent (i.e., inflation in Australia, IP A, is 5 percent), and the U.S. price index increases by 3 percent (i.e., inflation in the United States, IP US, is 3 percent). Calculate the new spot exchange rate of U.S. dollars for Australian dollars that should result from the differences in inflation rates. According to PPP, the 5 percent rise in the price of Australian goods relative to the 3 percent rise in the price of U.S. goods results in a depreciation of the Australian dollar (by 2 percent). Specifically, the exchange rate of Australian dollars to U.S. dollars should fall, so that: i US - i A = ΔS US\$/A\$ /S US\$/A\$ Plugging in the inflation and exchange rates, we get: = ΔS US\$/A\$ /S US\$/A\$ = ΔS US\$/A\$ /.759 or: -.02 = ΔS US\$/A\$ /.759 and: ΔS US\$/A\$ = -(.02).759 = Thus, it costs cents less to receive an Australian dollar (or it costs cents (75.9 cents cents), or of \$1, can be received for 1 Australian dollar). The Australian dollar depreciates in value by 2 percent against the U.S. dollar as a result of its higher inflation rate. 21. Assume that annual interest rates are 8 percent in the United States and 4 percent in Japan. An I can borrow (by issuing CDs) or lend (by purchasing CDs) at these rates. The spot rate is \$0.60/. 10

11 a. If the forward rate is \$0.64/, how could the I arbitrage using a sum of \$1million? What is the expected spread? Borrow \$1,000,000 in U.S. by issuing CDs Interest and principal at year-end = \$1,000,000 x 1.08 = \$1,080,000 Make a loan in Japan Interest and principal = \$1,000,000/0.60 = 1, x 1.04 = 1,733,333 Purchase U.S. dollars at the forward rate of \$0.64 x 1,733,333 = \$1,109, Spread = \$1,109, \$1,080,000 = \$29,333.33/1,000,000 = 2.93% b. What forward rate will prevent an arbitrage opportunity? The forward rate that will prevent any arbitrage is given by solving the following equation: (1+ r t = (1+ r D ust L jpt ) *St ) t = [( ) * 0.60]/(1.04) = \$0.6231/ 11

12 Chapter 14: X VAR 24. An I has \$100,000 of net positions outstanding in British pounds ( ) and -\$30,000 in Swiss francs (S). The standard deviation of the net positions as a result of exchange rate changes is 1 percent for the S and 1.3 percent for the. The correlation coefficient between the changes in exchange rates of the and the S is a. What is the risk exposure to the I of fluctuations in the /\$ rate? Since the I has a positive position, an appreciation of the will increase the value of its - denominated assets more than its liabilities, providing a net gain. The opposite will occur if the depreciates. b. What is the risk exposure to the I of fluctuations in the S/\$ rate? Since the I has a negative net position in Ss, the value of its Swiss-denominated assets will increase in value, but not as much as the value of its liabilities. Hence, an appreciation of the S will lead to a net loss. The opposite will occur if the currency depreciates. c. What is the risk exposure if both the and the S positions are aggregated? Use the DEAR formula for a portfolio: DEAR p= ( 100 ) ( ) + (-30 ) ( ) + 2( 0. 01)( )( 100 )(-30 )( 0. 8 ) = \$1, The I s net position is actually only \$1, Without including correlation, the exposure would have been estimated at \$100,000 - \$30,000 = \$70,

13 Chapter 22: Micro-Hedge 8. In each of the following cases, indicate whether it would be appropriate for an I to buy or sell a forward contract to hedge the appropriate risk. a. A commercial bank plans to issue CDs in three months. The bank should sell a forward contract to protect against an increase in interest rates. b. An insurance company plans to buy bonds in two months. The insurance company should buy a forward contract to protect against a decrease in interest rates. c. A savings bank is going to sell Treasury securities it holds in its investment portfolio next month. The savings bank should sell a forward contract to protect against an increase in interest rates. d. A U.S. bank lends to a rench company; the loan is payable in euros. The bank should sell francs forward to protect against a decrease in the value of the euro, or an increase in the value of the dollar. e. A finance company has assets with a duration of six years and liabilities with a duration of 13 years. The finance company should buy a forward contract to protect against decreasing interest rates that would cause the value of liabilities to increase more than the value of assets, thus causing a decrease in equity value. 13

14 9. The duration of a 20-year, 8 percent coupon Treasury bond selling at par is 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 /1.04 = years. b. What is the impact on the Treasury bond price if market interest rates increase 50 basis points? P = -MD( R)\$100,000 = x x \$100,000 = -\$4, 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= - MD( R) P= (0.005) \$95,000 = - \$4, 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? Decrease in market value of the bond purchase -\$4, Gain in value from the sale of futures contract \$4, Net gain or loss from hedge -\$ What are the differences between a microhedge and a macrohedge for a I? Why is it generally more efficient for Is to employ a macrohedge than a series of microhedges? 14

15 A microhedge uses a derivative contract such as a forward or futures contract to hedge the risk exposure of a specific transaction, while a macrohedge is a hedge of the duration gap of the entire balance sheet. Is that attempt to manage their risk exposure by hedging each balance sheet position will find that hedging is excessively costly, because the use of a series of microhedges ignores the I s internal hedges that are already on the balance sheet. That is, if a long-term fixed-rate asset position is exposed to interest rate increases, there may be a matching long-term fixed-rate liability position that also is exposed to interest rate decreases. Putting on two microhedges to reduce the risk exposures of each of these positions fails to recognize that the I has already hedged much of its risk by taking matched balance sheet positions. The efficiency of the macrohedge is that it focuses only on those mismatched positions that are candidates for off-balance-sheet hedging activities. 15

16 Chapter 22: Macro-Hedge 12. Hedge Row Bank has the following balance sheet (in millions): Assets \$150 Liabilities \$135 Equity 15 Total \$150 Total \$150 The duration of the assets is six years and the duration of the liabilities is four years. The bank is expecting interest rates to fall from 10 percent to 9 percent over the next year. a. What is the duration gap for Hedge Row Bank? DGAP = D A k D L = 6 (0.9)(4) = = 2.4 years b. What is the expected change in net worth for Hedge Row Bank if the forecast is accurate? Expected E = -DGAP[ R/(1 + R)]A = -2.4(-0.01/1.10)\$150m = \$3.272 million c. What will be the effect on net worth if interest rates increase 100 basis points? Expected E = -DGAP[ R/(1 + R)]A = -2.4(0.01/1.10)\$150 = -\$ d. If the existing interest rate on the liabilities is 6 percent, what will be the effect on net worth of a 1 percent increase in interest rates? 16

17 Solving for the impact on the change in equity under this assumption involves finding the impact of the change in interest rates on each side of the balance sheet, and then determining the difference in these values. The analysis is based on the equation: Expected E = A - L A = -D A [ R A /(1 + R A )]A = -6[0.01/1.10]\$150m = -\$ million and L = -D L [ R L /(1 + R L )]L = -4[0.01/1.06]\$135m = -\$ million Therefore, E = A - L = -\$8.1818m (-\$5.0943m) = - \$ million 16. Tree Row Bank has assets of \$150 million, liabilities of \$135 million, and equity of \$15 million. The asset duration is six years and the duration of the liabilities is four years. Market interest rates are 10 percent. Tree Row Bank wishes to hedge the balance sheet with Treasury bond futures contracts, which currently have a price quote of \$95 per \$100 face value for the benchmark 20-year, 8 percent coupon bond underlying the contract. Calculation of Duration for Problem 16 \$1,000 bond, 8% coupon, R = % and R = %, n = 20 years Cash Price = \$95 Time low PV of C PV of C x t

18 Total Duration = a. Should the bank go short or long on the futures contracts to establish the correct macrohedge? The bank should sell futures contracts since an increase in interest rates would cause the value of the equity and the futures contracts to decrease. But the bank could buy back the futures contracts to realize a gain to offset the decreased value of the equity. b. How many contracts are necessary to fully hedge the bank? If the market value of the underlying 20-year, 8 percent benchmark bond is \$95 per \$100, the market rate is percent (using a calculator) and the duration is as shown on the last page of this chapter solutions. The number of contracts to hedge the bank is: 18

19 ( DA kdl ) A (6 (0.9)4)\$150m N = = = 365contracts D x P x\$95,000 c. Verify that the change in the futures position will offset the change in the cash balance sheet position for a change in market interest rates of plus 100 basis points and minus 50 basis points. or an increase in rates of 100 basis points, the change in the cash balance sheet position is: Expected E = -DGAP[ R/(1 + R)]A = -2.4(0.01/1.10)\$150m = -\$3,272, The change in bond value = (0.01/ )\$95,000 = -\$9,079.41, and the change in 365 contracts is - \$9, x -365 = \$3,313, Since the futures contracts were sold, they could be repurchased for a gain of \$3,313, The sum of the two values is a net gain of \$41, or a decrease in rates of 50 basis points, the change in the cash balance sheet position is: Expected E = -DGAP[ R/(1 + R)]A = -2.4(-0.005/1.10)\$150m = \$1,636, The change in each bond value = (-0.005/ )\$95,000 = \$4, and the change in 365 contracts is \$4, x -365 = -\$1,656, Since the futures contracts were sold, they could be repurchased for a loss of \$1,656, The sum of the two values is a loss of \$20, d. If the bank had hedged with Treasury bill futures contracts that had a market value of \$98 per \$100 of face value, how many futures contracts would have been necessary to hedge fully the balance sheet? If Treasury bill futures contracts are used, the duration of the underlying asset is 0.25 years, the face value of the contract is \$1,000,000, and the number of contracts necessary to hedge the bank is: ( DA kdl ) A (6 (0.9)4)\$150m \$360,000,000 N = = = = 1, 469contracts D x P 0.25 x\$980,000 \$245,000 e. What additional issues should be considered by the bank in choosing between T-bond or T-bill futures contracts? 19

20 In cases where a large number of Treasury bonds are necessary to hedge the balance sheet with a macrohedge, the I may need to consider whether a sufficient number of deliverable Treasury bonds are available. The number of Treasury bill contracts necessary to hedge the balance sheet is greater than the number of Treasury bonds, the bill market is much deeper and the availability of sufficient deliverable securities should be less of a problem. 17. Reconsider Tree Row Bank in problem 16 but assume that the cost rate on the liabilities is 6 percent. a. How many contracts are necessary to fully hedge the bank? In this case, the bank faces different average interest rates on both sides of the balance sheet. urther, the yield on the bonds underlying the futures contracts is a third interest rate. Thus, the hedge also has the effects of basis risk. Determining the number of futures contracts necessary to hedge this balance sheet must consider separately the effect of a change in rates on each side of the balance sheet, and then consider the combined effect on equity. Estimating the number of contracts can be determined with the modified general equation shown on the next page. b. Verify that the change in the futures position will offset the change in the cash balance sheet position for a change in market interest rates of plus 100 basis points and minus 50 basis points. or an increase in rates of 100 basis points, E = 0.01[(4/1.06)\$135 m (6/1.10)\$150 m] = - \$3,087, The change in the bond value is (.01/ )\$95,000 = -\$9,079.41, and the change for -340 contracts = \$3,087, The sum of the two values is a net gain of \$ or a decrease in rates of 50 basis points, E = [(4/1.06)\$135 m (6/1.10)\$150 m] = \$1,543, The change in the bond value is (-.005/ )\$95,000 = \$4,539.71, and the change for -340 contracts = -\$1,543, The sum of the two values is a net loss of \$ Modified Equation Model for part (b): 20

21 { D ( N * P )* } = { D * * A D * * L } D ( N L A } + { + * R * L * R * A } { D ( N * P ) } + { (1+ R )*( MD * L MD * A) } { N * P } + { *( MD * L MD * A) } R (1+ R ) N R * P )* (1 + R ) = E = 0 + A L = 0 1 MD { MD * L + MD * A } P * MD { MD * A MD * L } A P * MD A D (1+ R ) L ( ) = = = ( \$818,181, \$509,433, ) \$907, \$308,747, = \$907, = 340contracts R (1+ R ) L 4 *150,000,000 *135,000, ,000* L A L L A A A L D (1+ R ) R (1+ R ) A = 0 L = = = c. If the bank had hedged with Treasury bill futures contracts that had a market value of \$98 per \$100 of face value (implying a discount rate of 8 percent), how many futures contracts would have been necessary to fully hedge the balance sheet? A market value of \$98 per \$100 of face value implies a discount rate of 8 percent on the underlying T-bills. Therefore, the equation developed above in part (a) to determine the number of contracts necessary to hedge the bank can be adjusted as follows for the use of T-bill contracts: 21

22 N = = ( MD * A MD * L ) A P * MD L ,000* 1.08 *150,000,000 ( ) ( \$818,181, \$509,433, ) = \$226, \$308,747, = \$226, = 1,361contracts *135,000, How would your answer for part (b) in problem 16 change if the relationship of the price sensitivity of futures contracts to the price sensitivity of underlying bonds were br = 0.92? The number of contracts necessary to hedge the bank would increase to 397 contracts. This can be found by dividing \$360,000,000 by ( x \$95,000 x 0.92). 21. Consider the following balance sheet (in millions) for an I: Assets Liabilities Duration = 10 years \$950 Duration = 2 years \$860 Equity \$90 a. What is the I's duration gap? The duration gap is 10 - (860/950)(2) = 8.19 years. 22

23 b. What is the I's interest rate risk exposure? The I is exposed to interest rate increases. The market value of equity will decrease if interest rates increase. c. How can the I use futures and forward contracts to put on a macrohedge? The I can hedge its interest rate risk by selling future or forward contracts. d. What is the impact on the I's equity value if the relative change in interest rates is an increase of 1 percent? That is, R/(1+R) = E = (950,000)(.01) = -\$77,800 e. Suppose that the I in part (c) macrohedges using Treasury bond futures that are currently priced at 96. What is the impact on the I's futures position if the relative change in all interest rates is an increase of 1 percent? That is, R/(1+R) = Assume that the deliverable Treasury bond has a duration of nine years. E = - 9(96,000)(.01) = -\$8,640 per futures contract. Since the macrohedge is a short hedge, this will be a profit of \$8,640 per contract. f. If the I wants to macrohedge, how many Treasury bond futures contracts does it need? To macrohedge, 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 = contracts. Rounding down, to construct a macrohedge requires the I 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? 23

24 In problem 21, we assumed that basis risk did not exist. That allowed us to assert that the percentage change in interest rates ( R/(1+R)) would be the same for both the futures and the underlying cash positions. If there is basis risk, then ( R/(1+R)) is not necessarily equal to ( R f /(1+R f )). If the I 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 [ R f /(1+R f ) / R/(1+R)] = br = 0.90 If br = 0.9, then: N f ( DA - k DL ) A 8.19(950,000) = = = 10 contracts D P br (9)(96,000)(.90) b. Explain what is meant by br = 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 ( 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 I must use more futures contracts to generate sufficient cash flows to offset the cash flows on its balance sheet position. 24. Village Bank has \$240 million worth of assets with a duration of 14 years and liabilities worth \$210 million with a duration of 4 years. In the interest of hedging interest rate risk, Village Bank is contemplating a macrohedge with interest rate T-bond futures contracts now selling for (32nds). The T-bond underlying the futures contract has a duration of nine years. If the spot and futures interest rates move together, how many futures contracts must Village Bank sell to fully hedge the balance sheet? 24

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