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Chapter 7 Managing interest rate risk using off-balance-sheet instruments Solutions for end-of-chapter questions Questions and problems What are derivative contracts? What is the value of derivative contracts to the managers of FIs? Which type of over-the-counter derivative contracts had the highest notional value outstanding globally as of December 2010? Derivatives are financial assets whose value is determined by the value of some underlying asset. As such, derivative contracts are instruments that provide the opportunity to take some action at a later date based on an agreement to do so at the current time. Although the contracts differ, the price, timing and extent of the later actions are usually agreed upon at the time the contracts are arranged. Normally, the contract values depend on the activity of the underlying asset. Derivative contracts have value to managers of FIs because of their ability to help in managing the various types of risk prevalent in the institutions. As of December 2010, the largest category of globally used over-the-counter derivatives was interest rate derivatives. What are some of the major differences between futures and forward contracts? How do these contracts differ from spot contracts? A spot contract is an exchange of cash or immediate payment, for financial assets or any other type of assets, at the time the agreement to transact business is made, that is, at time 0. Futures and forward contracts both are agreements between a buyer and a seller at time 0 to exchange the asset for cash (or some other type of payment) at a later time in the future. The specific grade and quantity of asset is identified at time 0, as is the specific price paid and time the transaction will eventually occur. One of the differences between futures and forward contracts is the uniqueness of forward contracts because they are negotiated between two parties. On the other hand, futures contracts are standardised because they are offered by and traded on an exchange. Futures contracts are marked to market daily by the exchange and the exchange guarantees the performance of the contract to both parties. Thus, the risk of default by either party is minimised from the viewpoint of the other party. No such guarantee exists for a forward contract. Finally, delivery of the asset almost always occurs for forward contracts, but seldom occurs for futures contracts. Instead, an offsetting or reverse transaction occurs through the exchange prior to the maturity of the contract. What is a naive hedge? How does a naive hedge protect an FI from risk? A hedge involves protecting the price of or return on an asset from adverse changes in price or return in the market. A naive hedge usually involves the use of a derivative instrument that has the same underlying asset as the asset being hedged. Thus, if a change in the price of the cash asset results in a gain, the same change in market value will cause the derivative instrument to generate a loss that offsets the gain in the cash asset. An FI holds a 15-year, $10 million par value bond that is priced at 104 with a yield to maturity of 7 per cent. The bond has a duration of eight years and the FI plans to sell it after two months. The FI’s market analyst predicts that interest rates will be 8 per cent at the time of the desired sale. Because most other analysts are predicting no change in rates, two-month forward contracts for 15-year bonds are available at 104. The FI would like to hedge against the expected change in interest rates with an appropriate position in a forward contract. What will this position be? Show that if rates rise 1 per cent as forecast, the hedge will protect the FI from loss. The expected change in the spot position is –8 ×$10 400 000 × (1/1.07) = –$777 570. This would mean a price change from 104 to 96.2243 per $100 face value of bonds. By entering into a two-month forward contract to sell $10 000 000 of 15-year bonds at 104, the FI will have hedged its spot position. If rates rise by 1 per cent and the bond value falls by $777 570, the FI can close out its forward position by receiving 104 for bonds that are now worth 96.2243 per $100 face value. The profit on the forward position will offset the loss in the spot market. The actual transaction to close the forward contract may involve buying the bonds in the market at 96.2243 and selling the bonds to the counterparty at 104 under the terms of the forward contract. Note that if a futures contract were used, closing the hedge position would involve buying a futures contract through the exchange with the same maturity date and dollar amount as the initial opening hedge contract. Contrast the position of being short with that of being long in futures contracts. To be short in futures contracts means that you have agreed to sell the underlying asset at a future time, while being long means that you have agreed to buy the asset at a later time. In each case, the price and the time of the future transaction are agreed upon when the contracts are initially negotiated. Suppose an FI purchases a T-bond futures contract at 95. (a) What is the FI’s obligation at the time the futures contract is purchased? You are obligated to take delivery of a $100 000 face value 20-year Treasury Bond at a price of $95 000 at some predetermined later date. (b) If an FI purchases this contract, in what kind of hedge is it engaged? This is a long hedge undertaken to protect the FI from falling interest rates. (c) Assume that the T-bond futures price falls to 94. What is the loss or gain? The FI will lose $1000 since the FI must pay $95 000 for bonds that have a market value of only $94 000. (d) Assume that the T-bond futures price rises to 97. Mark to market the position. In this case the FI gains $2000 since the FI pays only $95 000 for bonds that have a market value of $97 000. In each of the following cases, indicate whether it would be appropriate for an FI 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 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. The duration of a 20-year, 8 per cent coupon T-bond selling at par is 10.292 years. The bond’s interest is paid semi-annually and the bond qualifies for delivery against the T-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 T-bond price if market interest rates increase 50 basis points? P = –MD(R)$100 000 = –9.896 × 0.005 × $100 000 = –$4948.08. (c) If you sold a T-bond futures contract at 95 and interest rates rose 50 basis points, what would be the change in the value of your futures position? (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 –$4948.08 Gain in value from the sale of futures contract $4700.67 Net gain or loss from hedge –$247.41 What are the differences between a micro hedge and a macro hedge for an FI? Why is it generally more efficient for FIs to employ a macro hedge than a series of micro hedges? A micro hedge uses a derivative contract such as a forward or futures contract to hedge the risk exposure of a specific transaction, while a macro hedge is a hedge of the duration gap of the entire balance sheet. FIs 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 micro hedges ignores the FI’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 micro hedges to reduce the risk exposures of each of these positions fails to recognise that the FI has already hedged much of its risk by taking matched balance sheet positions. The efficiency of the macro hedge is that it focuses only on those mismatched positions that are candidates for off-balance-sheet hedging activities. What are the reasons why an FI may choose to hedge selectively its portfolio? Selective hedging involves an explicit attempt to not minimise the risk on the balance sheet. An FI may choose to hedge selectively in an attempt to improve profit performance by accepting some risk on the balance sheet or to arbitrage profits between a spot asset’s price movements and the price movements of the futures price. This latter situation often occurs because of differential changes in interest rates caused in part by cross-hedging. 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 per cent to 9 per cent over the next year. (a) What is the duration gap for Hedge Row Bank? DGAP = DA – k DL = 6 – (0.9)(4) = 6 – 3.6 = 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 = –$3.272. (d) If the existing interest rate on the liabilities is 6 per cent, what will be the effect on net worth of a 1 per cent increase in interest rates? 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 = –DA[RA/(1 + RA)]A = –6[0.01/1.10]$150m = –$8.1818 million and L = –DL[RL/(1 + RL)]L = –4[0.01/1.06]$135m = –$5.0943 million Therefore, E = A – L = –$8.1818m – (–$5.0943m) = – $3.0875 million For a given change in interest rates, why is the sensitivity of the price of a T-bond futures contract greater than the sensitivity of the price of a bank accepted bill futures contract? The price sensitivity of a futures contract depends on the duration of the asset underlying the contract. In the case of a bank accepted bill contract, the duration is 0.25 years. In the case of a T-bond contract, the duration is much longer. What is the meaning of the T-bond futures price bid of 95.75? A bid of 95.75 means that the buyer is happy to buy a futures contract on $100 000 bond with a yield to maturity of 4.25 per cent (i.e. 100 – 95.75). What is meant by fully hedging the balance sheet of an FI using futures contracts? Fully hedging the balance sheet involves using a sufficient number of futures contracts so that any loss (or gain) of net worth on the balance sheet is just offset by the gain (or loss) from the off-balance-sheet use of futures contracts for given changes in interest rates. 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 per cent. Tree Row Bank wishes to hedge the balance sheet with 20-year T-bond futures contracts, which are currently trading at 91.4705 (i.e. a yield of 8.5295%) and so have a price quote equivalent of $95 per $100 face value, 8 per cent coupon on the bond underlying the contract. Calculation of duration of the futures position $1000 bond, 8% coupon, R = 8.5295% and R = 8.2052%, n = 20 years Cash Price = $95 Time Flow PV of CF PV of CF × t 1 80 73.71268 73.71268 2 80 67.91950 135.83900 3 80 62.58161 187.74482 4 80 57.66323 230.65291 5 80 53.13139 265.65695 6 80 48.95572 293.73430 7 80 45.10822 315.75751 8 80 41.56301 332.50477 9 80 38.29659 344.66933 10 80 35.28681 352.86807 11 80 32.51357 357.64923 12 80 29.95828 359.49933 13 80 27.60381 358.84957 14 80 25.43439 356.08145 15 80 23.43546 351.53196 16 80 21.59364 345.49819 17 80 19.89656 338.24155 18 80 18.33286 329.99152 19 80 16.89206 320.94906 20 1080 210.12054 4202.41084 Total 950.00000 9853.84304 Duration = 10.3725 (a) Should the bank go short or long on the futures contracts to establish the correct macro hedge? 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 realise 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 per cent benchmark bond is $95 per $100, the market rate is 8.5295 per cent (using a calculator) and the duration is 10.3725 as shown on the last page of this chapter solutions. The number of contracts to hedge the bank is: (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. For 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 = –$3 272 727.27. The change in bond value = –10.3725(0.01/1.085295)$95 000 = –$9079.41 and the change in 365 contracts is –9079.41 × –365 = $3 313 986.25. Since the futures contracts were sold, they could be repurchased for a gain of $3 313 986.25. The sum of the two values is a net gain of $41 258.98. For 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 = $1 636 363.64. The change in each bond value = –10.37255(–0.005/1.085295)$95 000 = $4539.71 and the change in 365 contracts is $4539.71 × –365 = –$1 656 993.13. Since the futures contracts were sold, they could be repurchased for a loss of $1 656 993.13. The sum of the two values is a loss of $20 629.49. (d) If the bank had hedged with bank accepted 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: (e) What additional issues should be considered by the bank in choosing between T-bond or bank accepted bill futures contracts? In cases where a large number of Treasury Bonds are necessary to hedge the balance sheet with a macro hedge, the FI may need to consider whether a sufficient number of deliverable Treasury Bonds are available. The number of bank accepted bill contracts necessary to hedge the balance sheet is greater than the number of Treasury Bonds, the bill market is usually much deeper and the availability of sufficient deliverable securities should be less of a problem. What is basis risk? What are the sources of basis risk? Basis risk is the lack of perfect correlation between changes in the yields of the on-balance-sheet assets or liabilities and changes in interest rates on the futures contracts. The reason for this difference is that the cash assets and the futures contracts are traded in different markets. How would your answer for part (b) in problem 15 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 (10.3725 × $95 000 × 0.92). 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 macro hedge? 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 per cent? That is, R/(1 + R) = 0.01. E = – 8.19(950 000)(.01) = –$77 800 (e) Suppose that the FI in part (c) macro hedges using T-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 per cent? That is, R/(1 + R) = 0.01. Assume that the deliverable T-bond has a duration of nine years. E =– 9(96 000)(.01) = –$8640 per futures contract. Since the macro hedge is a short hedge, this will be a profit of $8640 per contract. (f) If the FI wants to macro hedge, how many T-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. Refer to problem 18. How does consideration of basis risk change your answers to problem 18? In problem 18, 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 (Rf/(1+Rf)). 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 macro hedge if [Rf/(1+Rf)/R/(1+R)] = br = 0.90 If br = 0.9, then: (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 per cent for every 1 per cent 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 macro hedge? 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. Assume an FI has assets of $250 million and liabilities of $200 million. The duration of the assets is six years and the duration of the liabilities is three years. The price of the futures contract is $115 000 and its duration is 5.5 years. (a) What number of futures contracts is needed to construct a perfect hedge if br = 1.10? (b) If Rf/(1 + Rf) = 0.0990, what is the expected R/(1 + R)? R/(1 + R) = (Rf/(1 + Rf))/br = 0.0990/1.10 = 0.09 How does using options differ from using forward or futures contracts? Both options and futures contracts are useful in managing risk. Other than the pure mechanics, the primary difference between these contracts lies in the requirement of what must be done on or before maturity. Futures and forward contracts require that the buyer or seller of the contracts must execute some transaction. The buyer of an option has the choice to execute the option or to let it expire without execution. The writer of an option must perform a transaction only if the buyer chooses to execute the option. What is a call option? What must happen to interest rates for the purchaser of a call option on a bond to make money? How does the writer of the call option make money? A call option is an instrument that allows the purchaser to buy some underlying asset at a pre-specified price on or before a specified maturity date. For example, the call option on a bond allows the owner to buy a bond at a specific price. For the owner of the option to make money, the option purchaser should be able to immediately sell the bond at a higher price. Thus, for the bond price to increase, interest rates must decrease between the time the option is purchased and the time it is executed. The writer of the call option makes a premium from the sale of the option. If the option is not exercised, the writer maximises profit in the amount of the premium. If the option is exercised, the writer stands to lose a portion or the entire premium and may lose additional money if the price on the underlying asset moves sufficiently far. What is a put option? What must happen to interest rates for the purchaser of a put option on a bond to make money? How does the writer of the put option make money? A put option is an instrument that allows the owner to sell some underlying asset at a pre-specified price on or before a specified maturity date. The put option on a bond allows the owner to sell a bond at a specific price. For the owner of the option to make money, the purchaser should be able to buy the bond at a lower price immediately prior to exercising the option. Thus, for the bond price to decrease, interest rates must increase between the time the option is purchased and the time it is executed. The writer of the put option makes a premium from the sale of the option. If the option is not exercised, the writer maximises profit in the amount of the premium. If the option is exercised, the writer stands to lose a portion or the entire premium and may lose additional money if the price on the underlying asset moves sufficiently far. Consider the following: (a) What are the two ways to use call and put options on T-bonds to generate positive cash flows when interest rates decline? Verify your answer with a diagram. First we must know whether the option is on price or yield as these move inversely. Assuming we are dealing with an option on price, the FI can either (a) buy a call option or (b) sell a put option on interest rate instruments, such as T-bonds, to generate positive cash flows in the event that interest rates decline. In the case of a call option, positive cash flows will increase as long as interest rates continue to decrease. See Figure 7.6 in the textbook as an example of positive cash flows minus the premium paid for the option. Although not labelled in this diagram, interest rates are assumed to be decreasing as you move from left to right on the X-axis. Thus, bond prices are increasing. The sale of a put option generates positive cash flows from the premium received. Figure 7.9 n the textbook shows that the payoff will decrease as the price of the bond falls. Of course, this can only happen if interest rates are increasing. Again, although not labelled in this diagram, interest rates are assumed to be increasing as you move from right to left on the X-axis. (b) Under what balance sheet conditions can an FI use options on T-bonds to hedge its assets and/or liabilities against interest rate declines? An FI can use call options on T-bonds to hedge an underlying cash position that decreases in value as interest rates decline. This would be true if, in the case of a macro hedge, the FI’s duration gap is negative and the repricing gap is positive. In the case of a micro hedge, the FI can hedge a single fixed-rate liability against interest rate declines. (c) Is it more appropriate for FIs to hedge against a decline in interest rates with long calls or short puts? An FI is better off purchasing calls as opposed to writing puts for two reasons. First, regulatory restrictions in some countries limit an FI’s ability to write naked short options. Second, since the potential positive cash inflow on the short put option is limited to the size of the put premium, there may be insufficient cash inflow in the event of interest rate declines to offset the losses in the underlying cash position. In each of the following cases, identify what risk the manager of an FI faces and whether the risk should be hedged by buying a put or a call option. (a) A bank plans to issue CDs in three months. The bank faces the risk that interest rates will increase. The FI should buy a put option. If rates rise, the CDs can be purchased at a lower price and sold immediately by exercising the option. The gain will offset the higher interest rate the FI must pay in the spot market. (b) An insurance company plans to buy bonds in two months. The insurance company (IC) is concerned that interest rates will fall and thus the price of the bonds will rise. The IC should buy call options that allow the bond purchase at the lower price. The bonds purchased with the options can be sold immediately for a gain that can be applied against the lower yield realised in the market. Or the bonds can be kept and placed in the IC’s portfolio if they are the desired type of asset. (c) A savings bank plans to sell Treasury securities next month. The savings bank is afraid that rates will rise and the value of the bonds will fall. The savings bank should buy a put option that allows the sale of the bonds at or near the current price. (d) A finance company has assets with a duration of six years and liabilities with a duration of 13 years. The FI is concerned that interest rates will fall, causing the value of the liabilities to rise more than the value of the assets, which would cause the value of the equity to decrease. Thus, the FI should buy a call option on interest rates (bonds). Consider an FI that wishes to use bond options to hedge the interest rate risk in the bond portfolio. (a) How does writing call options hedge the risk when interest rates decrease? In the case where the FI is long the bond, writing a call option will provide extra cash flow in the form of a premium. But falling interest rates will cause the value of the bond to increase and eventually the option will be exercised at a loss to the writer. But the loss is offset by the increase in value of the long bond. Thus, the initial goal of maintaining the interest rate return on the long bond can be realised. (b) Will writing call options fully hedge the risk when interest rates increase? Explain. Writing call options provides a premium that can be used to offset the losses in the bond portfolio caused by rising rates up to the amount of the premium. Further losses are not protected. (c) How does buying a put option reduce the losses on the bond portfolio when interest rates rise? When interest rates increase, the value of the bond falls, but the put allows the sale of the bond at or near the original price. Thus, the profit potential increases as interest rates continue to increase, although it is tempered by the amount of premium that was paid for the put. (d) Show by way of a diagram the purchase of a bond call option against the combination of a bond investment and the purchase of a bond put option. The profit payoff of a bond call option is given in Figure 7.6 in the textbook. If the price of the bond falls below the exercise price, the purchaser of the call loses the premium. As the price of the bond increases beyond the exercise price, the purchaser recovers the premium and then realises a net profit. Figures 7.11A and 7.11B in the textbook give the individual a net profit payoff of holding a bond long and the purchase of a put option. The put option allows a profit if bond prices drop. This profit will offset the loss on the long bond caused by the decrease in the bond value. If bond prices increase, the option will not be exercised and the investor will realise a gain from the increase in the bond’s value. Thus, the call option or the combination of long bond and put option give the same value. A pension fund manager anticipates the purchase of a 20-year, 8 per cent coupon T-bond at the end of two years. Interest rates are assumed to change only once every year at year end, with an equal probability of a 1 per cent increase or a 1 per cent decrease. The T-bond, when purchased in two years, will pay interest semi-annually. Currently, the T-bond is selling at par. (a) What is the pension fund manager’s interest rate risk exposure? The pension fund manager is exposed to interest rate declines (price increases). (b) How can the pension fund manager use options to hedge that interest rate risk exposure? This interest rate risk exposure can be hedged by buying call options on either financial securities or financial futures. (c) What prices are possible on the 20-year T-bonds at the end of year 1 and year 2? Currently, the bond is priced at par, $1000 per $1000 face value. At the end of the first year, either of two interest rates will occur. Either (a) interest rates will increase 1 per cent to 9 per cent (50 per cent probability of either occurrence) and the 20-year 8 per cent coupon Treasury Bond’s price will fall to $907.9921 per $1000 face value; or (b) interest rates will decrease 1 per cent to 7 per cent (50 per cent probability of occurrence). The 20-year 8 per cent coupon Treasury Bond’s price will increase to $1106.7754 per $1000 face value. At the end of two years, one of three different interest rate scenarios will occur: if (a) interest rates increase another 1 per cent to 10 per cent (25 per cent probability of occurrence), the 20-year 8 per cent coupon Treasury Bond’s price will fall to $828.4091 per $1000 face value; or if (b) interest rates decrease 1 per cent to 8 per cent or increase 1 per cent to 8 per cent (50 per cent probability of occurrence), the 20-year 8 per cent coupon Treasury Bond’s price will return to $1000 per $1000 face value; or if (c) interest rates decrease another 1 per cent to 6 per cent (25 per cent probability of occurrence), the 20-year 8 per cent coupon Treasury Bond’s price will increase to $1231.1477 per $1000 face value. (d) Show by way of diagram the prices over the two-year period. (e) If options on $100 000, 20-year, 8 per cent coupon T-bonds (both puts and calls) have a strike price of 101, what are the possible (intrinsic) values of the option position at the end of year 1 and year 2? The call option’s intrinsic value at the end of one year will be either: (a) zero if the price of a $100 000 20-year Treasury Bond is $90 799.21 (in the scenario that interest rates rise to 9 per cent); or (b) $110 677.54 – $101 000 (strike price) = $9677.54 if the price of a $100 000 20-year Treasury Bond is $110 677.54 (in the scenario that interest rates fall to 7 per cent). The call option’s intrinsic value at the end of two years will be either: (a) zero if the price of a $100 000 20-year Treasury Bond is $82 840.91 (in the scenario that interest rates rise to 10 per cent); or (b) zero if the price of a $100 000 20-year Treasury Bond is $100 000 in the scenario that interest rates stay at 8 per cent); or (c ) $123 114.77 – $101 000 (strike price) = $22 114.77 if the price of a $100 000 20-year Treasury Bond is $123 114.77 (in the scenario that interest rates fall to 6 per cent). (f) Show by way of a diagram the possible option values. (g) Using an 8 per cent discount factor calculate the option premium. PV = $9677.54/1.08 + $22 114.17/(1.08)2 = $10 773.25. Corporate Bank has $840 million of assets with a duration of 12 years and liabilities worth $720 million with a duration of seven years. The bank is concerned about preserving the value of its equity in the event of an increase in interest rates and is contemplating a macro hedge with interest rate options. The call and put options have a delta () of 0.4 and –0.4, respectively. The price of an underlying T-bond is 104.4, its coupon is 8 per cent and its modified duration is 7.6 years. (a) What type of option should Corporate Bank use for the macro hedge? The duration gap for the bank is [12 – (720/840)7] = 6. Therefore, the bank is concerned that interest rates may increase and it should purchase put options. As rates rise, the value of the bonds underlying the put options will fall, but they will be puttable at the higher put option exercise price. (b) How many options should be purchased? The bonds underlying the put options have a market value of $104 531.25. As they are 8 per cent bonds, the bonds are trading at a yield to maturity of 7.56 per cent. Given a modified duration of 7.6, the duration of these bonds is MD × 1.0756 = 8.17 years. Thus (c) What is the effect on the economic value of the equity if interest rates rise 50 basis points? Assuming R for the assets is similar to R for the underlying bonds, the change in equity value is: DGAP × A × (R/(1+R)) = –6($840 000 000)(.005/1.0756) = –$23 428 784. (d) What will be the effect on the hedge if interest rates rise 50 basis points? P = Np( × –MD × B × R) = 14 754 × –.4 × –7.6 × $104 531.25 × 0.005 = $23 442 262 gain (e) What will be the cost of the hedge if each option has a premium of $0.875? A price quote of $0.875 is per $100 face value of the put contract. Therefore, the cost per contract is $875 and the cost of the hedge is $875 × 14 754 = $12 909 750. (f) Show by way of a diagram the economic conditions of the hedge. The diagram of this portfolio position and the corresponding hedge is given in Figures 7.11A and B in the textbook. In this particular case, the payoff function for the net long position of the bank (DGAP = 6) should be considered as the payoff function of the bond in Figure 7.11A in the textbook. (g) How much must interest rates move against the hedge for the increased value of the bank to offset the cost of the hedge? Let E = $12 909 750 and solve the equation in part (c) above for R. Then R = $12 909 750 × 1.0756/($840 000 000×–6) = –0.002755 or –0.2755 per cent. (h) How much must interest rates move in favour of the hedge, or against the balance sheet, before the payoff from the hedge will exactly cover the cost of the hedge? Use the equation in part (d) above and solve for R. Then R = $12 909 750/[14 754 × –.4 × –7.6 × $104 531.25] = 0.002755 or 0.2755 per cent. (i) Formulate a management decision rule regarding the implementation of the hedge. If rates increase 0.2755 per cent, the equity will decrease in value approximately equal to the gain on the hedge. This position corresponds to the intersection of the payoff function from the put and the X-axis in Figure 7.11A in the textbook. The FI is out the cost of the hedge, which also will be the case for any other increase in interest rates. In effect, the cost of the hedge is the insurance premium to assure the value of the equity at the time the hedge is placed. If rates decrease approximately 0.2755 per cent, the gain on the equity value will offset the cost of the hedge and the put option will not be exercised. This position is shown by the intersection of the X-axis and the net payoff function in Figure 7.11B in the textbook. Any increase in rates beyond 0.2755 per cent will generate positive increases in value for the equity in excess of the cost of the hedge. An FI has a $200 million asset portfolio that has an average duration of 6.5 years. The average duration of its $160 million in liabilities is 4.5 years. The FI uses put options on T-bonds to hedge against unexpected interest rate increases. The average delta () of the put options has been estimated at –0.3 and the average duration of the T-bonds is seven years. The current market value of the T-bonds is $96 000. (a) What is the modified duration of the T-bonds if the current level of interest rates is 10 per cent? MD = D/1+.10 = 7/1.10 = 6.3636 years (b) How many put option contracts should it purchase to hedge its exposure against rising interest rates? The face value of the T-bonds is $100 000. = [6.5 – 4.5(.80)]× $200 000 000/[(–.3)×(–7.0)×(96 000)] = 2876.98 or 2877 contracts (c) If interest rates increase 50 basis points, what will be the change in value of the equity of the FI? Assuming R for the assets is similar to R for the underlying bonds, the change in equity value is: DGAP×A×(R/(1+R)) = –2.9($200 000 000)(.005/1.10) = –$2 636 363.64. (d) What will be the change in value of the T-bond option hedge position? P = Np( × –MD × B × R) = 2877 × –.3 × –6.3636 × $96 000 × 0.005 = $2 636 363.12 gain (e) If put options on T-bonds are selling at a premium of $1.25 per face value of $100, what is the total cost of hedging using options on T-bonds? Premium on the put options = 2877 x $1.25 x 1000 = $3 596 250. (f) What must be the change in interest rates before the change in value of the balance sheet (equity) will offset the cost of placing the hedge? Let E = $3 596 250 and solve the equation in part (c) above for R. Then R = $3 596 250 × 1.10/($200 000 000×–2.9) = –0.00682 or –0.68 per cent. (g) How much must interest rates change before the payoff of the hedge will exactly cover the cost of placing the hedge? Use the equation in part (d) above and solve for R. Then R = $3 596 250/[2877 × –.3 × –6.3636 × $96 000] = 0.00682 or 0.68 per cent. (h) Given your answer in part (f), what will be the net gain or loss to the FI? If rates decrease by 0.68 per cent, the increase in value of the equity will exactly offset the cost of placing the hedge. The options will be allowed to expire unused since the price of the bonds will be higher in the market place than the exercise price of the option. Explain the similarity between a swap and a forward contract. A forward contract requires delivery or taking delivery of some commodity or security at some specified time in the future at some price specified at the time of origination. In a swap, each party promises to deliver and/or receive a pre-specified series of payments at specific intervals over some specified time horizon. In this way, a swap can be considered to be the same as a series of forward contracts. Forwards, futures and option contracts had been used by FIs to hedge risk for many years before swaps were invented. If FIs already had these hedging instruments, why do they need swaps? Although similar in many ways, the following distinguishing characteristics cause the instruments to be differentiated: (a) The swap can be viewed as a portfolio of forward contracts with different maturity dates. Since cash flows on forward contracts are symmetric, the same can be said of swaps. This is in contrast to options, whose cash flows are asymmetric (truncated either on the positive or negative side depending upon the position). (b) Options are marked to market continuously, swaps are marked to market at coupon payment dates and forward contracts are settled only upon delivery (at maturity). Therefore, the credit risk exposure is greatest under a forward contract, where no third party guarantor exists as in options (the options clearing corporation for exchange-traded options) and swaps (the swap intermediary). (c) The transactions cost is highest for the option (the non-refundable option premium), next for the swap (the swap intermediary’s fee) and finally for the forward (which has no upfront payment). (d) Swaps also have a longer maturity than any other instrument and provide an additional opportunity for FIs to hedge longer term positions at lower cost. Moreover, since the package of forward contracts mirrors debt instruments, the swap provided FIs with a hedge instrument that is attractive and less costly than separate forward contracts. (e) Finally, the introduction of a swap intermediary reduces the credit risk exposure and the information and monitoring costs that are associated with a portfolio of individual forward contracts. Distinguish between a swap buyer and a swap seller. In which markets does each have the comparative advantage? The swap buyer makes the fixed-rate payments in an interest rate swap and the swap seller makes the variable-rate payments. This distinction occurs by convention. The notation in this text refers to the comparative advantage party as that which makes the specific swap payment. Thus, the buyer is said to have the comparative advantage in fixed-rate payments. Students will note that some other authors refer to the comparative advantage in the markets in which the cash financing occurs, which may not be the same market that would reduce the interest rate risk on the balance sheet and therefore the reason for the swap. Thus, in Example 7.6 in the textbook, the large national bank raises money in the fixed-rate market, even though the loans are variable-rate. An insurance company owns $50 million of floating-rate bonds yielding bank accepted bill rate (BBR) plus 1 per cent. These loans are financed by $50 million of fixed-rate guaranteed investment contracts (GICs) costing 10 per cent. A finance company has $50 million of car loans with a fixed rate of 14 per cent. The loans are financed by $50 million of CDs at a variable rate of BBR plus 4 per cent. (a) What is the risk exposure of the insurance company? The insurance company (IC) is exposed to falling interest rates on the asset side of the balance sheet. (b) What is the risk exposure of the finance company? The finance company (FC) is exposed to rising interest rates on the liability side of the balance sheet. (c) What would be the cash flow goals of each company if they were to enter into a swap arrangement? The IC wishes to convert the fixed-rate liabilities into variable-rate liabilities by swapping the fixed-rate payments for variable-rate payments. The FC wishes to convert variable-rate liabilities into fixed-rate liabilities by swapping the variable-rate payments for fixed-rate payments. (d) Which company would be the buyer and which company would be the seller in the swap? The FC will make fixed-rate payments and therefore is the buyer in the swap. The IC will make variable-rate payments and therefore is the seller in the swap. (e) Show by way of a diagram the direction of the relevant cash flows for the swap arrangement. Please see the diagram at below. Note that the fixed-rate swap payments from the finance company to the insurance company will offset the payments on the fixed-rate liabilities that the insurance company has incurred. The reverse situation occurs regarding the variable-rate swap payments from the insurance company to the finance company. Depending on the rates negotiated and the maturities of the assets and liabilities, both companies now have durations much closer to zero on this portion of their respective balance sheets. Note to instructors: it is useful to show on the diagram the cash-market financing cash flows when presenting the material on swaps. (f) What are reasonable cash flow amounts, or relative interest rates, for each of the payment streams? Determining a set of reasonable interest rates involves an analysis of the benefits to each firm. That is, does each firm pay lower interest rates than contractually obligated without the swap? Clearly, the direction of the cash flows will help reduce interest rate risk. One feasible swap is for the IC to pay the FC BBR + 2.5 per cent and for the FC to pay the IC 12 per cent. The net financing cost for each firm is given below. Finance company Insurance company Cash market liability rate BBR + 4% 10.0% Minus Swap-in rate –(BBR + 2.5%) –12.0% Plus Swap-out rate + 12% +(BBR + 2.5%) Net financing cost rate 13.5% BBR + 0.5% Whether the two firms would negotiate these rates depends on the relative negotiating power of each firm and the alternative rates for each firm in the alternative markets. That is, the fixed-rate liability market for the finance company and the variable-rate liability market for the insurance company. In a swap arrangement, the variable-rate swap cash flow streams often do not fully hedge the variable-rate cash flow streams from the balance sheet due to basis risk. (a) What are the possible sources of basis risk in an interest rate swap? First, the variable-rate index on the liabilities in the cash market may not match perfectly the variable-rate index negotiated into the swap agreement. This source of basis risk is similar to the cross-hedge risk in the use of futures contracts. Second, the premium over the index in the cash-market variable-rate liability may change over time as credit (default) risk conditions change. (b) How could the failure to achieve a perfect hedge be realised by the swap buyer? Swap pricing normally is based on a fixed notional amount over the life of the swap. If the fixed-rate asset portfolio of the buyer decreases over time, a fixed-notional amount swap agreement may not reflect accurately the desired interest rate risk goals of the buyer over the life of the swap. This situation could occur as loans are amortised (repaid in the normal context) or as prepayment rates change on either loans or bonds as macroeconomic conditions change. (c) How could the failure to achieve a perfect hedge be realised by the swap seller? The swap seller is subject to basis risk as discussed in part (a) above. A regional bank has $200 million of floating-rate loans yielding the BBR rate plus 2 per cent. These loans are financed by $200 million of fixed-rate deposits costing 9 per cent. A savings bank has $200 million of mortgages with a fixed rate of 13 per cent. They are financed by $200 million of CDs with a variable rate of the BBR rate plus 3 per cent. (a) Discuss the type of interest rate risk each FI faces. The regional bank is exposed to a decrease in rates that would lower interest income, while the savings bank is exposed to an increase in rates that would increase interest expense. In either case, profit performance would suffer. (b) One possible swap that would help both banks is the following: The regional bank sends variable-rate payments of the BBR rate + 1 per cent (T + 1%) to the savings bank and receives fixed-rate payments of 9 per cent from the savings bank. Show that this swap would be acceptable to both parties. Savings bank Major bank Cash market liability rate BBR + 3% 9.0% Minus swap-in rate –(BBR + 1%) –9.0% Plus swap-out rate + 9% +(BBR + 1%) Net financing cost rate 11.0% BBR + 1% The net interest yield on assets is 2 per cent (13% – 11%) for the savings bank and 1 per cent [(BBR + 2%) – (BBR + 1%)] for the major bank. An adjustment to make the net interest yield on assets equal at 1.5 per cent would be to have the savings bank pay a fixed rate of 9.5 per cent or receive a fixed rate of BBR + 0.5 per cent. Obviously, many rate combinations could be negotiated to achieve acceptable rate spreads and to achieve the desired interest rate risk management goals. (c) What are some of the practical difficulties in arranging this swap? The floating-rate assets may not be tied to the same rate as the floating-rate liabilities. This would result in basis risk. Also, if the mortgages are amortising, the interest payments would not match those on the notional amount of the swap. Bank 1 can issue five-year CDs at an annual rate of 11 per cent fixed or at a variable rate of BBR plus 2 per cent. Bank 2 can issue five-year CDs at an annual rate of 13 per cent fixed or at a variable rate of BBR plus 3 per cent. (a) Is a mutually beneficial swap possible between the two banks? A mutually beneficial swap exists because comparative advantage exists. (b) Where is the comparative advantage of the two banks? Bank 1 has a comparative advantage in the fixed-rate market because the difference in fixed rates is 2 per cent in favour of Bank 1. Bank 2 has the comparative advantage in the variable-rate market because the difference in variable rates is only –1 per cent against Bank 1. One way to compare the rate alternatives is to utilise the following matrix. Fixed rate Variable rate Bank 1 11.0% BBR + 2% Bank 2 13.0% BBR + 3% Difference –2.0% –1% (c) What is the net quality spread? The net quality spread is the difference between the fixed-rate versus variable-rate differential. Thus, the net quality spread = –2% – (–1%) = –1%. This amount represents the net amount of gains (interest savings) to be allocated between the firms. (d) What is an example of a feasible swap? Many rate combinations are possible to achieve the quality spread or reduced interest charge of 1 per cent. The following is a framework to achieve the outside boundaries of acceptable interest rates using the matrix of possible rates shown in part (b). Using the rates shown for Bank 1, the negotiated swap rates will give the entire quality spread to Bank 2. The diagram and payoff matrix below verifies this case. The relative payoffs are given below: Bank 2 Bank 1 Cash market liability rate BBR + 3% 11.0% Minus swap-in rate –(BBR + 2%) –11.0% Plus swap-out rate + 1% +(BBR + 2%) Net financing cost rate 12.0% BBR + 2% Bank 1 is paying the rate it could achieve in the variable-rate market, thus Bank 1 receives no benefit from these swap rates. Now consider the rates shown for Bank 2 in the matrix of rates in part (b). In this case, Bank 2 is receiving the exact rate it owes on the liabilities and it is paying the rate necessary if it was in the fixed-rate market. Bank 1 receives the entire 1 per cent benefit as it is paying net 1 per cent less than it would need to pay in the variable-rate market. The relative payoffs are given below: Bank 2 Bank 1 Cash market liability rate BBR + 3% 11.0% Minus swap-in rate –(BBR + 2%) –13.0% Plus swap-out rate + 11% +(BBR + 3%) Net financing cost rate 12.0% BBR + 1% Any swap rate combination between these two boundaries that yields a total saving of 1 per cent in combined interest cost becomes a feasible set of negotiated swap rates. The exact set of rates will depend on the negotiating position of each bank and the expected interest rates over the life of the swap. As an example, consider the average of the two fixed-rate payments and the average of the two variable-rate payments. The relative payoffs are given below: Bank 2 Bank 1 Cash market liability rate BBR + 3% 11.0% Minus swap-in rate –(BBR + 2.5%) –12.0% Plus swap-out rate + 12% +(BBR + 2.5%) Net financing cost rate 12.5% BBR + 1.5% In each case, the banks are paying 0.5 per cent less than they would in the relative desired cash markets. First Bank can issue one-year, floating-rate CDs at BBR plus 1 per cent or fixed-rate CDs at 12.5 per cent. Second Bank can issue one-year, floating-rate CDs at BBR plus 0.5 per cent or fixed-rate at 11 per cent. (a) What is a feasible swap with all of the benefits going to First Bank? The possible interest rate alternatives faced by each bank are given below: Fixed rate Variable rate Bank 1 12.5% BBR + 1.0% Bank 2 11.0% BBR + 0.5% Difference –1.5% 0.5% The quality spread is 1.5% – 0.5% = 1.0%. Second Bank has the comparative advantage in the fixed-rate market and First Bank has the comparative advantage in the variable-rate market. A set of swap rates within the feasible boundaries that will give all the benefits to First Bank is 11 per cent fixed rate and BBR + 0.5 per cent variable rate. (b) What is a feasible swap with all of the benefits going to Second Bank? A set of rates within the feasible boundaries that will give all the benefits to Second Bank is 12.5 per cent fixed rate and BBR + 1.0 per cent variable rate. (c) Show by way of a diagram each situation. Show by way of a diagram all of the benefits going to First Bank. The payoff matrix that demonstrates that all of the benefits go to First Bank follows. First Bank Second Bank Cash market liability rate BBR + 1% 11.0% Minus swap-in rate –(BBR + 0.5%) –11.0% Plus swap-out rate + 11% +(BBR + 0.5%) Net financing cost rate 11.5% BBR + 0.5% The net cost for First Bank is 11.5 per cent or 1 per cent less than it would pay in the fixed-rate cash market. The net cost for Second Bank is exactly the same as it would pay in the variable-rate cash market. Show by way of a diagram all of the benefits going to Second Bank. The net cost for First Bank is 12.5 per cent, which is exactly what it would pay in the fixed-rate cash market. The net cost for Second Bank is BBR–0.5 per cent or 1 per cent less than it would pay in the variable-rate cash market. The payoff matrix that illustrates that all of the benefits go to Second Bank follows. First Bank Second Bank Cash market liability rate BBR + 1% 11.0% Minus swap-in rate –(BBR + 1%) –12.5% Plus swap-out rate + 12.5% +(BBR + 1%) Net financing cost rate 12.5% BBR–0.5% (d) What factors will determine the final swap arrangement? The primary factor that will determine the final distribution of the swap rates is the present value of the cash flows for the two parties. The most important no-arbitrage condition is that the present value of the expected cash flows made by the buyer should equal the present value of the expected cash flows made by the seller. Secondary factors include the negotiating strengths of either party to the transaction. Two multinational corporations enter their respective debt markets to issue $100 million of two-year notes. Firm A can borrow at a fixed annual rate of 11 per cent or a floating rate of LIBOR plus 50 basis points, repriced at the end of the year. Firm B can borrow at a fixed annual rate of 10 per cent or a floating rate of LIBOR, repriced at the end of the year. (a) If Firm A is a positive duration gap insurance company and Firm B is a money market mutual fund, in what market(s) should each firm borrow so as to reduce their interest rate risk exposures? Firm A will prefer to borrow in the fixed-rate debt market in order to generate positive cash flows when interest rates increase. This will offset the impact of an increase in interest rates, which would cause the market value of the insurance company’s equity to decline. Firm B will prefer to borrow in the floating-rate debt market so as to better match the duration of its short-term assets. (b) In which debt market does Firm A have a comparative advantage over Firm B? The matrix of possible interest rates is given below. Fixed rate Variable rate Bank 1 11.0% L + 0.5% Bank 2 10.0% L % Difference –1.0% 0.5% Firm A has a comparative advantage in the floating-rate market and Firm B has a comparative advantage in the fixed-rate market. This is because the default risk premium of Firm A over Firm B is 50 basis points in the floating-rate market and 100 basis points in the fixed-rate market. (c) Although Firm A is riskier than Firm B and therefore must pay a higher rate in both the fixed-rate and floating-rate markets, there are possible gains to trade. Set up a swap to exploit Firm A’s comparative advantage over Firm B. What are the total gains from the swap trade? Assume a swap intermediary fee of 10 basis points. The total gains to the swap trade are 50 basis points (the price differential on Firm A’s default risk premium over Firm B) less 10 basis points (the swap intermediary fee). Both Firms A and B can exploit this price differential by issuing in the debt market in which they have comparative advantage and then swapping the interest payments. The 40 basis points can be allocated to either Firm A and/or Firm B according to the terms of the swap. A possible set of feasible swap rates that give all of the gains to Firm A (see part (d) below) is illustrated here. Evidence that Firm A receives all of the benefits is given in the payoff matrix below. Firm A Firm B Cash market liability rate L + 0.5% 10.0% Minus swap-in rate –(L %) –10.0% Plus swap-out rate + 10.0% +(L %) Net financing cost rate 10.5% L % Less intermediary fee 0.1% Financing cost rate net of fee 10.6% Firm A is paying the intermediary fee, since Firm B is receiving no benefits from this swap transaction. The 40 basis point net differential could be shared in a number of other combinations where Firm A received most (exploited) of the benefit. (d) The gains from the swap trade can be apportioned between Firm A and B through negotiation. What terms of trade would give all the gains to Firm A? What terms of trade would give all the gains to Firm B? All the gains go to Firm A if Firm B pays LIBOR for Firm A’s floating-rate debt. Then Firm A must pay 10 per cent for Firm B’s fixed-rate debt plus 50 basis points on Firm A’s floating-rate debt plus 10 basis points for the swap intermediary’s fee. The total fixed annual interest cost to Firm A is 10.6 per cent, a savings of 40 basis points over the cash-market fixed rate of 11 per cent. This swap rate apportionment is illustrated in part (c) above. All the gains go to Firm B if Firm A pays 11 per cent for Firm B’s fixed-rate, 10 per cent debt. Then Firm B pays LIBOR plus 50 basis points on Firm A’s floating-rate debt for a net savings of 50 basis points. The savings occur because Firm B receives an excess 1.0 per cent from Firm A, but must pay 50 basis points more to Firm A than it would pay in the cash floating-rate market. Firm A must pay 11 per cent against Firm B’s fixed-rate debt, but receives its exact liability payment from Firm B. A diagram of this allocation is given below. In this example, Firm B would pay the swap intermediary fee of 10 basis points and thus would realise a net, after-fee savings of 40 basis points. The payoff matrix is given below. Firm A Firm B Cash market liability rate L + 0.5% 10.0% Minus swap-in rate –(L + 0.5%) –11.0% Plus swap-out rate + 11.0% +(L + 0.5 %) Net financing cost rate 11.0% L–0.5% Less intermediary fee 0.1% Financing cost rate net of fee L–0.4% (e) Assume swap pricing that allocates all the gains from the swap to Firm A. If A buys the swap from B and pays the swap intermediary’s fee, what are the end-of-year net cash flows if LIBOR is 8.25 per cent? Firm A (in millions of dollars) Firm B A pays out fixed rate ($10.00) B pays out LIBOR ($8.25) A receives LIBOR from B $8.25 B receives fixed rate from A $10.00 A pays floating-rate to creditors (L + 0.5%) ($8.75) B pays fixed-rate to creditors (10.00) A pays intermediary fee ($0.10) A’s net cash inflow ($10.60) B’s net cash inflow ($8.25) This solution is an extension of the diagram in part (c) and the explanation at the beginning of part (d) above where LIBOR is 8.25 per cent. The summary shows the effective cost rate converted to dollars for the total cash flows of each firm. However, the cash flows in a swap arrangement include only the differential cash flows between the two parties. Thus at end of year, Firm A would pay $1.75 ($10.00 – $8.25) to Firm B and $0.10 to the intermediary for a total cash flow on the swap arrangement of $1.85. Firm B receives $1.75 from Firm A. (f) If Firm A buys the swap in part (e) from Firm B and pays the swap intermediary’s fee, what are the end-of-year net cash flows if LIBOR is 11 per cent? Be sure to net swap payments against cash market payments for both firms. Firm A (in millions of dollars) Firm B A pays out fixed rate ($10.00) B pays out LIBOR ($11.00) A receives LIBOR from B $11.00 B receives fixed rate from A $10.00 A pays floating-rate to creditors (L + 0.5%) ($11.50) B pays fixed-rate to creditors (10.00) A pays intermediary fee ($0.10) A’s net cash inflow ($10.60) B’s net cash inflow ($11.00) Even though LIBOR has increased to 11 per cent, Firm A’s total effective cost rate has not changed. The rate remains at 10.60 per cent or a total of $10.60 million. However, the cost rate for Firm B has increased because LIBOR has increased. Thus, the actual cash flows in the swap transaction now become that Firm B pays $1.00 ($11 – $10) to Firm A and that Firm A receives $1.00 and pays out $0.10 to the intermediary. Each firm must, of course, pay the cash market liability rates. (g) If all barriers to entry and pricing inefficiencies between Firm A’s debt markets and Firm B’s debt markets were eliminated, how would that affect the swap transaction? If relative prices are the same in the markets of both Firm A and B, then there are no potential gains to trade and therefore no swap transactions can take place. They will each issue debt in their own markets. An FI has $500 million of assets with a duration of nine years and $450 million of liabilities with a duration of three years. The FI wants to hedge its duration gap with a swap that has fixed-rate payments with a duration of six years and floating-rate payments with a duration of two years. What is the optimal amount of the swap to effectively macro hedge against the adverse effect of a change in interest rates on the value of the FI’s equity? Using the formula: NS = [(DA – kDL)A]/(DFixed – DFloating) = [(9 – 0.9 × 3)$500 million]/(6 – 2) = $787.5 million. Web questions 40 Go to the website of the Australian Securities Exchange (www.asx.com.au) and find the latest quotes for the 90-day bank accepted bill futures and options contracts. The answer will depend on the date of the assignment. At the Australian Securities Exchange website, click on ‘Futures & Options’. Then click on ‘Interest Rate Futures & Options’, then click on ‘More about Interest Rate Futures & Options’. I'm unable to browse the internet or access specific websites like the Australian Securities Exchange. However, you can easily find the latest quotes for the 90-day bank accepted bill futures and options contracts by following these steps: 1. Visit the Australian Securities Exchange website: www.asx.com.au 2. Navigate to the "Prices and research" section, which may be located in the main menu or under a specific tab related to trading or financial products. 3. Look for the section dedicated to futures and options contracts. This may be labeled as "Futures & Options" or something similar. 4. Within the futures and options section, locate the relevant market for bank accepted bill futures and options contracts. This market may be categorized under interest rate derivatives or a similar category. 5. Once you've found the market for bank accepted bill futures and options contracts, you should be able to view the latest quotes, including prices, volumes, and other relevant information for these contracts. If you encounter any difficulties or have trouble finding the information, you may also consider contacting the Australian Securities Exchange directly or referring to their help or support resources for assistance. 41 View and read the CNBC article by, K. Holliday, 'Rising Interest Rates Next Big Risk for Asia: World Bank', published 12 Jun 2013, found at www.cnbc.com/id/100811757 . If you were the risk manager of an Asian Bank, what strategies could you use to counter the potential risk associated with the predicted changes in interest rates highlighted by the article. The suggestion by the world bank is that Asian banks should be aware that there is more chance that interest rates could rise than fall. In such environments, an FI could reduce its risk by matching the duration of its asset portfolio and liability portfolio. However, if this is not possible through physical products, the FI manager could use futures and/or options to immunize the balance sheet of the bank from interest rate rises. Essentially, the risk manager would want to ensure that the duration of the liability portfolio was longer than the duration of the asset portfolio so that the bank can earn higher yields from its assets, than the additional cost of liabilities brought about by the higher interest rate. Integrated mini case: hedging interest rate risk with futures versus options On 4 January 2012, an FI has the following balance sheet (rates = 10 per cent) Assets Liabilities/equity A 200m DA = 6 years L 170m DL = 4 years E 30m DGAP = [6 – (170/200)4] = 2.6 years > 0 The FI manager thinks rates will increase by 0.75 per cent in the next three months. If this happens, the equity value will change by: The FI manager will hedge this interest rate risk with either futures contracts or option contracts. If the FI uses futures, it will select June T-bonds to hedge. The duration on the T-bonds underlying the contract is 14.5 years and the T-bonds are selling at a price per $100 000 m or $114 343.75. T-bond futures rates, currently 9 per cent, are expected to increase by 1.25 per cent over the next three months. If the FI uses options, it will buy puts on 15-year T-bonds with a June maturity, an exercise price of 113 and an option premium of 1.5625 per cent. The spot price on the T-bond underlying the option is 135.71875 per cent. The duration on the T-bonds underlying the options is 14.5 years and the delta of the put options is –0.75. Managers expect these T-bond rates to increase by 1.24 per cent from 7.875 per cent in the next three months. If by 4 April 2012, balance sheet rates increase by 0.8 per cent, futures rates by 1.4 per cent and T-bond rates underlying the option contract by 0.95 per cent, would the FI have been better off using the futures contract or the option contract as its hedge instrument? For the hedge with futures contracts: contracts On 4 April 2012, as the FI gets out of the futures hedge: Loss on balance sheet Gain off balance sheet (futures) The net gain is $3 970 909 – $3 781 818 = $189 091 For a hedge with option contracts: contracts On 4 April 2012, as the FI gets out of the option hedge: Loss on balance sheet Gain off balance sheet (options) = = –$3 781 818 $3 621 701 The net gain is $3 621 701 – $3 781 818 = –$160 117 In this case, the FI would be better off hedging with futures contracts rather than option contracts. If, by 4 April 2012, balance sheet rates actually fall by 0.75 per cent, futures rates fall by 1.05 per cent and T-bond rates underlying the option contract fall by 1.24 per cent, would the FI have been better off using the futures contract or the option contract as its hedge instrument? For the hedge with futures contracts: contracts On 4 April 2012, as the FI gets out of the futures hedge: Loss on balance sheet Gain off balance sheet (futures) The net gain is $3 545 454 – $2 978 182 = $567 272 For a hedge with option contracts: , contracts On 4 April 2012, as the FI gets out of the option hedge, the value of the T-bond underlying the put option has increased. The FI does not have to exercise these options if the loss on exercise is greater than the option premium. Thus: Loss on balance sheet Gain off balance sheet (options) Exercise: = = $3 545 454 –$4 727 272 No exercise: O = 278.63821100 000(–1 36/64%) = –$435 372 The FI will not exercise the options, taking the loss. Rather, it will let the options expire unused. Thus, the net gain is $3 545 454 – $435 372 = $3 110 082 In this case, the FI would be much better off hedging with option contracts rather than futures contracts. Solution Manual for Financial Institutions Management Anthony Saunders, Marcia Cornett, Patricia McGraw 9780070979796, 9780071051590