CH-27-28 Interest Rate Options – Bond Markets, Analysis and Strategies : Book Guide

This Document Contains Chapters 27 to 28 CHAPTER 27 INTEREST-RATE OPTIONS CHAPTER SUMMARY In this chapter we explain the various types of interest-rate options, their applications to portfolio management, and how they are priced. OPTIONS DEFINED An option is a contract in which the writer of the option grants the buyer of the option the right to purchase from or sell to the writer a designated instrument at a specified price within a specified period of time. The writer, also referred to as the seller, grants this right to the buyer in exchange for a certain sum of money called the option price or option premium. The price at which the instrument may be bought or sold is called the strike or exercise price. The date after which an option is void is called the expiration date. An American option may be exercised at any time up to and including the expiration date. A European option may be exercised only on the expiration date. When an option grants the buyer the right to purchase the designated instrument from the writer or (seller), it is called a call option. When the option buyer has the right to sell the designated instrument to the writer, the option is called a put option. The buyer of any option is said to be long the option; the writer is said to be short the option. DIFFERENCES BETWEEN AN OPTION AND A FUTURES CONTRACT Notice that options differ from futures contracts, in that the buyer of an option has the right but not the obligation to perform, whereas the option seller (writer) has the obligation to perform. In the case of a futures contract, both the buyer and the seller are obligated to perform. TYPES OF INTEREST-RATE OPTIONS Interest-rate options can be written on cash instruments or futures. At one time, there were several exchange-traded option contracts whose underlying instrument was a debt instrument. Exchange-Traded Futures Options An option on a futures contract, commonly referred to as a futures option, gives the buyer the right to buy from or sell to the writer a designated futures contract at a designated price at any time during the life of the option. Mechanics of Trading Futures Options Upon exercise, the futures price for the futures contract will be set equal to the exercise price. The position of the two parties is then immediately marked to market in terms of the then-current futures price. Thus the futures position of the two parties will be at the prevailing futures price. At the same time, the option buyer will receive from the option seller the economic benefit from exercising. Because the writer or (seller) of an option has agreed to accept all of the risk (and none of the reward) of the position in the underlying instrument, the writer is required to deposit not only the margin required on the interest rate futures contract position if that is the underlying instrument, but also (with certain exceptions) the option price that is received for writing the option. In addition, as prices adversely affect the writer’s position, the writer would be required to deposit variation margin as it is marked to market. There are three reasons why futures options on fixed-income securities have largely supplanted options on physicals as the options vehicle of choice for institutional investors. First, unlike options on fixed-income securities, options on Treasury coupon futures do not require payments for accrued interest to be made. Second, futures options are believed to be “cleaner” instruments because of the reduced likelihood of delivery squeezes. Finally, in order to price any option, it is imperative to know at all times the price of the underlying instrument. Specifications for the Actively Traded Futures Options There are options on all of the futures contracts. All futures options are of the American type. Trading of futures options on Treasury bonds stops in the month prior to the underlying futures contract’s delivery month. In an attempt to compete with the over-the-counter (OTC) option market, flexible Treasury futures options were introduced. These futures options allow counterparties to customize options within certain limits. INTRINSIC VALUE AND TIME VALUE OF AN OPTION The cost to the buyer of an option is primarily a reflection of the option’s intrinsic value and any additional amount over that value. The premium over intrinsic value is often referred to as time value. Intrinsic Value of an Option The intrinsic value of an option is the economic value of the option if it is exercised immediately. The intrinsic value of a call option on a bond is the difference between the bond price and the strike price. When a call option has intrinsic value, it is said to be in-the-money. When the strike price of a call option exceeds the bond price, the call option is said to be out-of-the-money and has no intrinsic value. An option for which the strike price is equal to the current bond price is said to be at-the-money. For a put option, the intrinsic value is equal to the amount by which the bond price is below the strike price. Time Value of an Option The time value of an option is the amount by which the option price exceeds the intrinsic value. There are two ways in which an option buyer may realize the value of a position taken in the option. First, the investor may exercise the option. Secondly, the investor may sell the call option. The latter alternative is never less than the first and always greater if the time value is not equal to zero. PROFIT AND LOSS PROFILES FOR SIMPLE NAKED OPTION STRATEGIES To appreciate the opportunities available with interest-rate options, the profit and loss profiles for various option strategies must be understood. We begin with simple strategies in only one option on a bond, which are referred to as naked option strategies. A naked option strategy means that no other position is taken in another option or bond. The four naked option strategies that we illustrate are (i) long call strategy (buying call options), (ii) short call strategy (selling or writing call options), (iii) long put strategy (buying put options), and (iv) short put strategy (selling or writing put options). Long Call Strategy (Buying Call Options) The most straightforward option strategy for participating in an anticipated decrease in interest rates (increase in the price of bonds) is to buy a call option on a debt instrument. This is called a long call strategy. To illustrate this strategy, suppose that the current price of an 8% coupon paying bond is $100, which is its par value), which means that the yield on this bond is currently 8%. As the strike price is equal to the current price of the bond, this option is at-the-money. The profit and loss from the strategy will depend on the price of the bond at the expiration date. The price, in turn, will depend on the yield on 20-year bonds with an 8% coupon, because in one month the bond will have only 20 years to maturity. We can use a hypothetical call option to demonstrate the speculative appeal of options. Suppose that an investor has strong expectations that market yields will fall in one month. With an option price of $4, the speculator can purchase 25 call options for each $100 invested. Thus if the market yield declines, the investor realizes the price appreciation associated with 25 bonds of $100 par each (or $2,500 par). With the same $100, the investor could buy only one $100 par value bond and realize the appreciation associated with that one bond if the market yield declines. Now, suppose that in one month the market yield declines to 6% so that the price of the bond increases to $123.11. The long call strategy will result in a profit of [($23.11)(25)] – $100 = $477.75, which is a return (less transaction costs) of $477.75 / $100 = 4.7775 or 477.75% on the $100 investment in the call options. The long bond strategy results merely in a profit of $23.11, a 23.11% return on $100. It is this greater leverage that an option buyer can achieve that attracts investors to options when they wish to speculate on interest-rate movements. It does not come without drawbacks, however. Suppose that the market yield is unchanged at the expiration date so that the price of the bond is $100. The long call strategy will result in the loss of the entire investment of $100, whereas the long bond strategy will produce neither a gain nor a loss. Short Call Strategy (Selling or Writing Call Options) An investor who believes that interest rates will rise or change very little can, if those expectations prove correct, realize income by writing (selling) a call option. This strategy is called a short call strategy. Long Put Strategy (Buying Put Options) The most straightforward option strategy for benefiting from an expected increase in interest rates is to buy a put option. This strategy is called a long put strategy. Whereas an investor who pursues a short bond strategy participates in all the upside potential and faces all the downside risk, the long put strategy allows the investor to limit the downside risk to the option price while still maintaining upside potential. However, the upside potential is less than that for a short put position by an amount equal to the option price. Short Put Strategy (Selling or Writing Put Options) The short put strategy involves the selling (writing) of put options. This strategy is employed if the investor expects interest rates to fall or stay flat so that the price of the bond will increase or stay the same. The profit and loss profile for a short put option is the mirror image of that for the long put option. The maximum loss is limited only by how low the price of the bond can fall by the expiration date less the option price received for writing the option. Considering the Time Value of Money Our illustrations of the four naked option positions do not reflect the time value of money. Specifically, the buyer of an option must pay the seller the option price at the time the option is purchased. The seller, in contrast, assuming that the option price does not have to be used as margin for the short position, has the opportunity to invest this option price. We also ignored the time value of money in comparing the option strategies with positions in the underlying instrument. Reinvestment income must be factored into the analysis of an option position. Also, the effects of financing costs and opportunity costs on the long or short bond positions, respectively, must be factored into the analysis. PUT-CALL PARITY RELATIONSHIP AND EQUIVALENT POSITIONS The put-call parity relationship is the relationship between the price of a call option and the price of a put option on the same underlying instrument, with the same strike price and the same expiration date. Ignoring the time value of money and considering European options, the outcome from the following position must be one of no arbitrage profits: long the bond + short call option + long put option = 0. In terms of price, it can be shown that there will be no arbitrage profits at any time (not just expiration) if Ppo = Pco + S – Pb where Ppo = price of put option, Pco = price of call option, S = strike price of option, Pb = current price of the underlying bond, and the strike price and expiration date are the same for both options. Considering the time value of money, the put-call parity relationship for coupon bonds is Ppo = Pco + PV(S) + PV(coupon) – Pb where PV(S) = present value of the strike price, and PV(coupon) = present value of the coupon payments. Equivalent Positions Working with equation of long the bond + short call option + long put option = 0, we can identify equivalent positions; that is, positions that will provide the same profit profile. For example, subtracting the long put position from both sides of the above equation, we have long the bond + short call option = –long put option But the position on the right-hand side of equation (28.4) is the same as a short put position. Thus, long the bond + short call option = short put option The latter equation is a covered call position, which is a long bond position plus a short call option position on the same bond. It has the same profit profile as a short put option position. OPTION PRICE Six factors will influence the option price: (i) current price of the underlying instrument; (ii) strike price; (iii) time to expiration; (iv) short-term risk-free interest rate over the life of the option; (v) coupon rate on the bond; and (vi) expected volatility of yields (or prices) over the life of the option. The impact of each of these factors may depend on whether the option is a call or a put, the option is an American option or a European option, and the underlying instrument is a bond or a futures contract on a bond. MODELS FOR PRICING OPTIONS Several models have been developed for determining the theoretical value of an option. These models are referred to as option pricing models. There are models for valuing options on bonds and options on bond futures. Models for Valuing Options on Bonds The most popular model is the Black-Scholes option pricing model. The option price derived from the Black-Scholes option pricing model is “fair” in the sense that if any other price existed, it would be possible to earn riskless arbitrage profits by taking an offsetting position in the underlying stock. There are three assumptions underlying the Black-Scholes model that limit its use in pricing options on interest-rate instruments. First, the probability distribution for the return assumed by the Black-Scholes option pricing model permits some probability—no matter how small—that the return can take on any positive value. The second assumption of the Black-Scholes option pricing model is that the short-term interest rate is constant over the life of the option. The third assumption is that the variance of prices is constant over the life of the option. While the Black-Scholes model is applicable to many securities (most noteworthy common stock), the model is limited in pricing options on bonds. The proper way to value options on interest-rate instruments is to use an arbitrage-free model that takes into account the yield curve. These models can incorporate different volatility assumptions along the yield curve. A binomial interest-rate tree can be used to value a stand-alone European option on a bond. The arbitrage-free binomial model satisfies the put-call parity relationship for European. Models for Valuing Options on Bond Futures The most commonly used model for futures options was developed by Fischer Black. The model was initially developed for valuing European options on forward contracts. There are two problems with this model. First, the Black model does not overcome the problems cited earlier for the Black-Scholes model. Failing to recognize the yield curve means that there will not be a consistency between pricing Treasury futures and options on Treasury futures. Second, the Black model was developed for pricing European options on futures contracts. Treasury futures options, however, are American options. The second problem can be overcome. Despite the many limitations and inconsistent assumptions of the Black model for valuing futures options, it has been widely adopted by traders for computing the implied volatility from options on Treasury bond futures options. These implied volatilities are also published by some investment houses and are available through data vendors. When computing implied volatilities of yield from Treasury bond futures options, the process is more complex than those for options on individual stocks or stock indexes. Remember that the options are written on futures prices. Therefore, the implied volatilities computed directly from the Black model are implied price volatilities of the underlying futures contract. Selection and Interpretation of Implied Volatility The implied volatility for both in-the-money and out-of-the money options with the same expiration date are higher than the implied volatility for at-the-money options with the same expiration date. The U-shaped curve that exists if the strike price is plotted on the horizontal axis and the implied volatility on the vertical axis looks like a smile, the relationship is referred to as volatility smile. More recently, however, the pattern that has been observed between the strike price and implied volatility indicates that the implied volatility decreases with the strike price, a pattern referred to as volatility skew. Standard practice suggests that the implied volatility of the at-the-money or the nearest-the money option should be used. SENSITIVITY OF OPTION PRICE TO CHANGE IN FACTORS In employing options in an investment strategy, a portfolio manager would like to know how sensitive the price of an option is to a change in any one of the factors that affect its price. Call Option Price and Price of the Underlying Bond Because the theoretical call option price is shown by the convex line, the difference between the theoretical call option price and the intrinsic value at any given price for the underlying bond is the time value of the option. The slope of the tangent line shows how the theoretical call option price will change for small changes in the price of the underlying bond. The slope is popularly referred to as the delta of the option. Specifically, delta = . Call Option Price and Time to Expiration All other factors constant, the longer the time to expiration, the greater the option price. Because each day the option moves closer to the expiration date, the time to expiration decreases. The theta of an option measures the change in the option price as the time to expiration decreases, or equivalently, it is a measure of time decay. Theta is measured as follows: theta = . Call Option Price and Expected Interest Rate Volatility All other factors constant, a change in the expected interest rate volatility will change the option price. The kappa of an option measures the dollar price change in the price of the option for a 1% change in the expected price volatility. That is, kappa = . Duration of an Option The modified duration of an option measures the price sensitivity of the option to changes in interest rates. The modified duration of an option can be shown to be equal to modified duration for an option = As expected, the modified duration of an option depends on the modified duration of the underlying bond. It also depends on the price responsiveness of the option to a change in the underlying instrument. HEDGE STRATEGIES Hedge strategies involve taking a position in an option and a position in the underlying bond in such a way that changes in the value of one position will offset any unfavorable price (interest rate) movement in the other position. Two popular hedge strategies are: (i) the protective put buying strategy, and (ii) the covered call writing strategy. Hedging Long-Term Bonds with Puts on Futures Investors often want to hedge their bond positions against a possible increase in interest rates. Buying puts on futures is one of the easiest ways to purchase protection against rising rates. The process is not complicated. It involves simply (i) the relationship between price and yield, (ii) the assumed relationship between the yield spread between the hedged bonds and the cheapest-to-deliver issue, and (iii) the conversion factor for the cheapest-to-deliver issue. Covered Call Writing with Futures Options Unlike the protective put strategy, covered call writing is not entered into with the sole purpose of protecting a portfolio against rising rates. The covered call writer, believing that the market will not trade much higher or much lower than its present level, sells out-of-the-money calls against an existing bond portfolio. Comparing Alternative Strategies Three basic hedging strategies for hedging a bond position are: (i) hedging with futures, (ii) hedging with out-of-the-money protective puts, and (iii) covered call writing with out-of-the-money calls. Similar but opposite strategies exist for those who want to avoid the risk caused by decreases in interest rates. As might be expected, there is no “best” strategy. Except for the perfect hedge, there is always some range of possible final values of the portfolio. Of course, exactly what that range is, and the probabilities associated with each possible outcome, is a matter of opinion. KEY POINTS An option grants the buyer of the option the right either to buy (in the case of a call option) or to sell (in the case of a put option) the underlying asset to the seller (writer) of the option at the strike (exercise) price by the expiration date. The option price or option premium is the amount that the option buyer pays to the writer of the option. An American option allows the option buyer to exercise the option at any time up to and including the expiration date; a European option may be exercised only at the expiration date. Interest-rate options include options on fixed-income securities and options on interest-rate futures contracts, called futures option. The latter are the preferred vehicle for implementing investment strategies. Because of the difficulties of hedging particular bond issues or pass-through securities, many institutions find over-the-counter options more useful; these contracts can be customized to meet specific investment goals. The buyer of an option cannot realize a loss greater than the option price and has all the upside potential. By contrast, the maximum gain that the writer (seller) of an option can realize is the option price; the writer is exposed to all the downside risk. The option price consists of two components: the intrinsic value and the time value. The intrinsic value is the economic value of the option if it is exercised immediately (except that if there is no positive economic value that will result from exercising immediately, the intrinsic value is zero). The time value is the amount by which the option price exceeds the intrinsic value. Six factors influence the option price: (1) the current price of the underlying bond, (2) the strike price of the option, (3) the time remaining to the expiration of the option, (4) the expected price volatility of the underlying bond (i.e., expected interest-rate volatility), (5) the short-term risk-free interest rate over the life of the option, and (6) coupon payments. An option pricing model determines the theoretical or fair value of an option. There are option pricing models for options on bonds (i.e., options on physicals) and options on bond futures. The two models used to value options on bonds are the Black–Scholes option pricing model and the arbitrage-free binomial option pricing model. The limitations of the former when applied to options on bonds are that it fails to incorporate the yield curve in the model and does not recognize that there is a maximum price that a bond can reach. The most common model to value an option on a bond futures contract is the Black model. The two popular strategies using options are protective put buying and covered call writing. ANSWERS TO QUESTIONS FOR CHAPTER 27 (Questions are in bold print followed by answers.) 1. An investor owns a call option on bond × with a strike price of 100. The coupon rate on bond × is 9% and has 10 years to maturity. The call option expires today at a time when bond × is selling to yield 8%. Should the investor exercise the call option? To exercise this call option it must be in-the-money, i.e., the current price (Pb) must be greater than the strike price (S) of $100. We use the bond valuation formula as given below to find the current price: Pb = C + where C = semiannual payment of 0.045(100) = $4.5, r is the current yield of 8% / 2 = 4% or 0.04, n is the number of periods which is 2(10) = 20, and M is the par value of $100. Inserting in our values, we have: Pb = $4.5 + = $4.5 + = $4.5 + $100(0.4563869) = $4.5(13.590326) + $45.63869 = $61.15647 + $45.63869 = $106.79516 or about $106.80. Because the option is a call option and the strike price of $100 is less than the current market value of $106.80, the investor should exercise before it expires today and buy Bond × at $100 and make a profit of $6.80. If the investor doesn’t actually buy the bond (which is often the case), then the investor and seller of the call option will cancel their positions with the investor collecting $6.80. Either way there is $6.80 profit per $100 value of Bond X. 2. When the buyer of a put option on a futures contract exercises, explain the resulting position for the buyer and the writer. The buyer of a put option has a long position on a futures contract exercises when the strike price is greater than the current price of the futures contract (this is the opposite found for a call option). The buyer’s position is said to be in-the-money position (i.e., the exercise or intrinsic value is positive). However, to make a net profit the exercise value must be greater that the cost paid for the contract. The writer or seller has a short position and is obligated to pay the buyer of the option the exercise value of the put option contract. This position for the writer is not total loss because the writer has collected the contract price for selling the option. The writer may even make a profit if the exercise value is less than the contract price in which case the exercise value does not overcome the original purchase price of the option contract. As the parties to the futures option will realize a position in a futures contract when the option is exercised, the question is: What will the futures price be? That is, at what price will the long be required to pay for the instrument underlying the futures contract, and at what price will the short be required to sell the instrument underlying the futures contract? Upon exercise, the futures price for the futures contract will be set equal to the exercise price. The position of the two parties is then immediately marked to market in terms of the then-current futures price. Thus the futures position of the two parties will be at the prevailing futures price. At the same time, the option buyer will receive from the option seller the economic benefit from exercising. In the case of a put futures option, the option writer must pay the option buyer the difference between the exercise price and the current futures price. 3. An investor wants to protect against a rise in the market yield on a Treasury bond. Should the investor purchase a put option or a call option to obtain protection? An investor wanting to protect against a rise in the market yield of a Treasury bond is worried that its current claim to an asset will fall in value as yields increase. Thus, the investor should purchase a put option thereby locking in selling price in case the asset falls in value. For example, if the investor owns a bond priced at $100, the value will fall if interest rates take off. Suppose the bond value falls to $90, the strike price is $98, and cost is $1. The position in the put option has reduced the loss to ($100 – $98) – $1 = $3 instead of ($100 – $90) = $10. The net gain is $3 – ($10) = $7. 4. What is the intrinsic value and time value of a call option on bond W given the following information? strike price of call option = 97 current price of bond W = 102 call option price = 9 The intrinsic value for a call option is the current price minus the strike price. Thus, the intrinsic value = 102 – 97 = 5 or $5 per $100. The time premium is the difference between the current market price for the call option and the intrinsic value. Thus, time premium or time value = 9 – 5 = 4 or $4 per $100. 5. “There’s no real difference between options and futures. Both are hedging tools, and both are derivative products. It’s just that with options you have to pay an option premium, whereas futures require no upfront payment except for a ‘good faith’ margin. I can’t understand why anyone would use options.” Do you agree with this statement? One would not agree with the statement because an option, unlike a futures, does not lose substantial value. For example, consider a call option where the buyer’s profit potential is unlimited but its loss is only the cost of the option. Such an outcome cannot be achieved using futures. Thus, one could understand why an investor would choose to use options even though it does require an upfront cost. More details are given below. From the buyer’s point of view, the only cost with an option is the price of the option and that is the maximum amount that can be lost. With a futures contract much more can be lost plus the additional costs related to maintaining a margin account. With a futures contract, we have a contract that is marked to market, which means money can be spent even before the contract’s maturity. This cost can be much greater than the cost of an option. Furthermore, options differ from futures contracts, in that the buyer of an option has the right but not the obligation to perform, whereas the option seller (writer) has the obligation to perform. In the case of a futures contract, both the buyer and the seller are obligated to perform. Also in a futures contract, the buyer does not pay the seller to accept the obligation; in the case of an option, the buyer pays the seller the option price. The risk/reward characteristics of the two contracts are also different. In a futures contract the long position realizes a dollar-for-dollar gain when the price of the futures increases and suffers a dollar-for-dollar loss when the price of the futures decreases. The opposite occurs for the short position. Options do not provide such a symmetric risk/reward relationship. The most that a long option position can lose is the option price; yet the long retains all the upside potential, although the gain is always reduced by the option price. The maximum profit that the short option position may realize is the option price, but this position has substantial downside risk. 6. What arguments would be given by those who feel that the Black-Scholes model does not apply in pricing interest-rate options? The arguments lie in its underlying assumptions. There are three assumptions underlying the Black-Scholes model that limit its use in pricing options on interest-rate instruments. First, the probability distribution for the return assumed by the Black-Scholes option pricing model permits some probability—no matter how small—that the return can take on any positive value. But in the case of a zero-coupon bond, the price cannot take on a value above $100 and therefore the return is capped. In the case of a coupon bond, we know that the price cannot exceed the sum of the coupon payments plus the maturity value. For example, for a five-year 10% coupon bond with a maturity value of $100, the price cannot be greater than $150 (five coupon payments of $10 plus the maturity value of $100). Thus, unlike stock returns, bond prices have a maximum return. The only way that a bond’s return can exceed the maximum value is if negative interest rates are permitted. This is not likely to occur, so any probability distribution for prices assumed by an option pricing model that permits bond prices to be higher than the maximum bond value could generate nonsensical option prices. The Black-Scholes model does allow bond prices to exceed the maximum bond value (or, equivalently, allows negative interest rates). That is one of the reasons why we can get a senseless option price for the three-month European call option on the three-year zero-coupon bond using the Black-Scholes model. The second assumption of the Black-Scholes option pricing model is that the short-term interest rate is constant over the life of the option. Yet the price of an interest-rate option will change as interest rates change. A change in the short-term interest rate changes the rates along the yield curve. Therefore, to assume that the short-term rate will be constant is inappropriate for interest-rate options. The third assumption is that the variance of prices is constant over the life of the option. However, as a bond moves closer to maturity its price volatility declines. Therefore, the assumption that price variance is constant over the life of the option is inappropriate. 7. Below are some excerpts from an article titled “It’s Boom Time for Bond Options as Interest-Rate Hedges Bloom,” published in the November 8, 1990, issue of The Wall Street Journal. Answer each question given after the below quotes. (a) “The threat of a large interest-rate swing in either direction is driving people to options to hedge their portfolios of long-term Treasury bonds and medium-term Treasury notes,” said Steven Northern, who manages fixed-income mutual funds for Massachusetts Financial Services Co. in Boston. Why would a large interest rate swing in either direction encourage people to hedge? A large interest rate swing in either direction (compared to one direction) could double the demand for derivatives such as options. This is because regardless of one’s long or short position in one’s assets (i.e., portfolio of long-term Treasury bonds and medium-term Treasury notes), one would be concerned about a loss in value. In the case of Mr. Northern whose firm owns bonds, they may plan on selling some of their portfolio in the future or may even have contracts to deliver some bonds to other parties. They would be concerned that increases in interest rates would cause the value to fall before the sale. Mr. Northern’s firm may also be buying bonds for their portfolio in the future (for example, to replace those that are maturing or simply to invest incoming funds). They would be concerned that decreases in interest rates would cause the value to increase before the purchase. The extent of Northern’s concern is a function not only of the expected volatility (either way) in interest rates but also the maturity. Thus, there would be more concern with long-term Treasury bonds than with medium-term Treasury notes. (b) “If the market moves against an option purchaser, the option expires worthless, and all the investor has lost is the relatively low purchase price, or ‘premium,’ of the option.” Comment on the accuracy of this statement. This statement is essentially true because with an option one has the right but not the obligation to exercise and also the costs are much less than acquiring the underlying asset. However, there is also the opportunity cost involved because the option price could have been reinvested in less risky investments and locked in a guaranteed return. Thus, when the option expires worthless, a more accurate estimate of the loss would be the option price plus any earned interest. Finally, options can be expensive if the underlying asset is subject to volatility. This is because the buyer’s cost is proportionate to the probability of the underlying asset changing in value. (c) “Futures contracts also can be used to hedge portfolios, but they cost more, and there isn’t any limit on the amount of losses they could produce before an investor bails out.” Comment on the accuracy of this statement. Futures contracts, in terms of dealer fees, are a relatively cheap form of hedging and do not cost much. Most investors in futures contract are not worried about losing on the futures because they have the opposite position in the underlying asset. In brief, futures contracts are often used as a form of insurance with any lost in the futures contract canceled out by the underlying asset for which futures contracts are used to hedge. However, if one is a speculator and takes a position in the futures market without owning the underlying asset then losses could indeed become quite large. (d) “Mr. Northern said Massachusetts Financial has been trading actively in bond and note put options. ‘The concept is simple,’ he said. ‘If you’re concerned about interest rates but don’t want to alter the nature of what you own in a fixed-income portfolio, you can just buy puts.’ ” Why might put options be a preferable means of altering the nature of a fixed-income portfolio? When managing a fixed-income portfolio, an alternative to buying put options is to short sale an asset. This will be more expensive because one would have to (i) eventually invest an amount similar to what is already invested in the current asset that is owned or (ii) be willing to sell an asset currently found in your portfolio to cover your short position. It is simpler and more economical to simply buy put options as a way of altering the nature of a fixed-income portfolio. That way, one will lock in a price that will preserve the desired value. 8. What are the differences between an option on a bond and an option on a bond futures contract? A major difference concerns the underlying asset. A bond is usually part of a portfolio whose value you want to protect. A bond futures is usually acquired to hedge the underlying asset that one owns. There would also likely be a difference in terms of the current price. The option on a bond entitles the purchaser to cash in when the market price of the bond relative to strike value is favorable. An option on a bond futures contract entitles the purchaser to cash in when the futures price of the bond relative to strike value is favorable. Thus, this difference involves the strike price relative to the designated market price which in one case is the bond’s market price and in the other case the bond’s futures price. Whereas, the difference in these two prices could be substantial, they are more likely to be similar. 9. What is the motivation for the purchase of an over-the-counter option? Over-the-counter (or dealer) options typically are purchased by institutional investors who are motivated to hedge the risk associated with a specific security. For example, a thrift may be interested in hedging its position in a specific mortgage pass-through security. Typically, the maturity of the option coincides with the time period over which the buyer of the option wants to hedge, so the buyer is usually not concerned with the option’s liquidity. Interest-rate options include options on fixed-income securities and options on interest-rate futures contracts. The latter, more commonly called futures options, are the preferred vehicle for implementing investment strategies. However, because of the difficulties of hedging particular bond issues or pass-through securities, many institutions are motivated to use over-the-counter options because they are more useful in that they can be customized to meet specific interest-rate futures contracts. Finally, there are over-the-counter options on the shape of the yield curve or the yield spread between two securities that investors may find desirable and useful. These options include those on the spread between mortgage pass-through securities and Treasuries, or between double A corporates and Treasuries. 10. Does it make sense for an investor who wants to speculate on interest-rate movements to purchase an over-the-counter option? It doesn’t always make sense for an investor who wants to simply speculate on interest-rate movements to purchase an over-the-counter option. This is especially true if the tailor-made option costs more. In brief, there are plenty of standard contracts that are less expensive and can give one profit by correctly speculating on interest rates movements. Also, as described below, there are flexible Treasury futures options that allow customization subject to constraints. In an attempt to compete with the over-the-counter (OTC) option market, flexible Treasury futures options were introduced. These futures options allow counterparties to customize options within certain limits. Specifically, the strike price, expiration date, and type of exercise (American or European) can be customized subject to constraints. One key constraint is that the expiration date of a flexible contract cannot exceed that of the longest standard option. Unlike an OTC option, where the option buyer is exposed to counterparty risk, a flexible Treasury futures option is guaranteed by the clearing house. The minimum size requirement for the launching of a flexible futures option is 100 contracts. 11. “I don’t understand how portfolio managers can calculate the duration of an interest-rate option. Don’t they mean the amount of time remaining to the expiration date?” Respond to this question. Duration is not designed to measure the time remaining to expiration but to measure the price sensitivity to changes in interest rates. The length of time to the expiration date of the option is positively related to the option value but is not the same. More details are given below. The duration of an option measures the price sensitivity of the option to changes in interest rates. The modified duration of an option can be shown to be equal to the modified duration of underlying instrument times delta times the price of underlying instrument divided by the price of the option. Delta is the change in the price of a call option divided by the change in the price of the underlying asset. For example, a delta of 0.4 means that a $1 change in the price of the underlying asset will change the price of the call option by approximately $0.40. As expected, the modified duration of an option depends on the modified duration of the underlying bond. It also depends on the price responsiveness of the option to a change in the underlying instrument, as measured by the option’s delta. The leverage created by a position in an option comes from the last ratio in the formula. The higher the price of the underlying instrument relative to the price of the option, the greater the leverage (i.e., the more exposure to interest rates for a given dollar investment). It is the interaction of all three factors that affects the modified duration of an option. For example, a deep-out-of-the-money option offers higher leverage than a deep-in-the-money option, but the delta of the former is less than that of the latter. Because the delta of a call option is positive, the modified duration of an interest rate call option will be positive. Thus, when interest rates decline, the value of an interest rate call option will rise. A put option, however, has a delta that is negative. Thus the modified duration is negative. Consequently, when interest rates rise, the value of a put option rises. 12. Answer the below questions. (a) What factors affect the modified duration of an interest-rate option? The modified duration of an interest-rate option is influenced by the modified duration of the underlying bond, delta (the price responsiveness of the option to a change in the underlying instrument), and the leverage. The higher the price of the underlying instrument relative to the price of the option, the greater the leverage (i.e., the more exposure to interest rates for a given dollar investment). (b) Deep-in-the-money option always provides a higher modified duration for an option than a deep-out-of-the-money option. Comment. The modified duration of an option can be shown to be equal to the modified duration of underlying instrument times delta times the price of underlying instrument divided by the price of the option. Delta is the change in the price of a call option divided by the change in the price of the underlying asset. The leverage created by a position in an option comes from the last ratio in the formula (i.e., from the price of underlying instrument divided by the price of the option). While a deep-in-the-money option has a delta that is greater than a deep-out-of-the-money option, its leverage is less. Thus, it is not necessarily true that a deep-in-the-money option always provides a higher modified duration for an option than a deep-out-of-the-money option. (c) The modified duration of all options is positive. Is this statement correct? The modified duration of an interest-rate option is influenced by the modified duration of the underlying bond (which is positive), delta (which can be positive or negative), and the leverage (which is positive). The higher the price of the underlying instrument relative to the price of the option, the greater the leverage (i.e., the more exposure to interest rates for a given dollar investment). The delta is the price responsiveness of the option to a change in the underlying instrument and is positive for a call option. Thus, the modified duration of an interest rate call option will be positive (i.e., when interest rates decline, the value of an interest rate call option will rise). However, a put option has a delta that is negative. Thus, its modified duration is negative (i.e., when interest rates rise, the value of a put option rises). Thus, the statement is incorrect because the modified duration of all options is not positive. 13. How is the implied volatility of an option determined? Six factors will influence the option price: (i) current price of the underlying instrument; (ii) strike price; (iii) time to expiration; (iv) short-term risk-free interest rate over the life of the option; (v) coupon rate on the bond; and (vi) expected volatility of yields (or prices) over the life of the option. The impact of each of these factors may depend on whether the option is a call or a put, the option is an American option or a European option, and the underlying instrument is a bond or a futures contract on a bond. The only one of these factors that is not known and must be estimated is the expected volatility of yield (i.e., expected volatility of price over the life of the option). A popular methodology to assess whether an option is fairly priced is to assume that the option is priced correctly and then, using an option pricing model, estimate the volatility that is implied by that model, given the observed option price and the other five factors that determine the price of an option. The estimated volatility computed in this manner is called the implied volatility. For example, suppose that a portfolio manager using some option pricing model, the current price of the option, and the five other factors that determine the price of an option computes an implied yield volatility of 12%. If the portfolio manager expects that the volatility of yields over the life of the option will be greater than the implied volatility of 12%, the option is considered to be undervalued. In contrast, if the portfolio manager’s expected volatility of yields over the life of the option is less than the implied volatility, the option is considered to be overvalued. Although we have focused on the option price, the key to understanding the options market is knowing that trading and investment strategies in this market involve buying and selling volatility. Estimating the implied volatility and comparing it with the trader’s or portfolio manager’s expectations of future volatility is just another way of evaluating options. If an investor uses expected volatility to compute the fair value of the option, the option will appear cheap or expensive in exactly the same cases. Despite the many limitations and inconsistent assumptions of the Black model for valuing futures options, it has been widely adopted by traders for computing the implied volatility from options on Treasury bond futures options. These implied volatilities are also published by some investment houses and are available through data vendors. When computing implied volatilities of yield from Treasury bond futures options, the process is more complex than those for options on individual stocks or stock indexes. Remember that the options are written on futures prices. Therefore, the implied volatilities computed directly from the Black model are implied price volatilities of the underlying futures contract. How do we interpret the meaning of implied volatility? For example, what is the meaning of an “implied yield volatility of 0.91%”? To interpret this number, one needs to be aware that this number is extracted from the observed option price based on the Black model. As a result, the meaning of this number not only depends on the assumption that the market correctly prices the option, but also the fact that the market prices the option in accordance with the Black model. Neither of these assumptions needs to hold. In fact, most probably, both assumptions are unrealistic. Given these assumptions, one may interpret that the option market expects a constant annualized yield volatility of 0.91% for 30-year Treasury from April 30, 1997, to the maturity date of the option. 14. What are the delta and gamma of an option? Suppose we are considering how the price of the call option changes when the price of the underlying bond changes. In graphing this relationship, the slope of the tangent line shows how the theoretical call option price will change for small changes in the price of the underlying bond. The slope is popularly referred to as the delta of the option. Specifically, delta = . For example, a delta of 0.4 means that a $1 change in the price of the underlying bond will change the price of the call option by approximately $0.40. The curvature of the convex relationship can also be approximated. This is the rate of change of delta as the price of the underlying bond changes. The measure is commonly referred to as gamma and is defined as follows: gamma = . 15. Explain why the writer of an option would prefer an option with a high theta (all other factors equal). The loss to the writer decreases to the extent the option price declines. Thus, the writer of an option would prefer a high theta (all other factors equal) because a high theta means that the option price declines quickly as it moves toward the expiration date. More details are given below. All other factors constant, the longer the time to expiration, the greater the option price. Because each day the option moves closer to the expiration date, the time to expiration decreases. The theta of an option measures the change in the option price as the time to expiration decreases, or equivalently, it is a measure of time decay. Theta is measured as follows: theta = . Assuming that the price of the underlying bond does not change (which means that the intrinsic value of the option does not change), theta measures how quickly the time value of the option changes as the option moves towards expiration. Buyers of options prefer a low theta so that the option price does not decline quickly as it moves toward the expiration date. An option writer benefits from an option that has a high theta. 16. In implementing a protective put buying strategy, explain the trade-off between the cost of the strategy and the strike price selected. The trade-off is that the cost of the strategy increases as a higher strike price is selected. More details are given below. The cost of a put option increases as the strike price increases. Thus, for options issued with high strike prices, the cost of the put option will be high. The reason for this is that the probability of exercising the put option (and also making more profit) increases when the strike price is set high. This explains why you can find put options identical in all respects except for the strike price, and the put option with the higher strike price will sell at a higher price. 17. Here is an excerpt from an article titled “Dominguez Barry Looks at Covered Calls,” appearing in the July 20, 1992, issue of Derivatives Week, p. 7: SBC Dominguez Barry Funds Management in Sydney, with A$5.5 billion under management, is considering writing covered calls on its Australian bond portfolio to take advantage of very high implied volatilities, according to Carl Hanich, portfolio manager. The implied price volatility on at-the-money calls is 9.8%, as high as Hanich can ever remember… In response to rising volatility, Hanich is thinking about selling calls with a strike of 8.5%, generating premium income. “I’d be happy to lose bonds at 8.5%, given our market’s at 8.87% now,” he said. Explain the strategy that Mr. Hanich is considering. A covered call position, which is a long bond position plus a short call option position on the same bond, has the same profit profile as a short put option position. Thus, when Hanich states he is willing to lose (or sell) bonds at 8.5%, he is suggesting that he would use a put option strike price consistent with an 8.5% bond. One advantage (of Hanich’s strategy) is that if bond prices don’t fall but remain the same, then he makes a profit on the sell of the call option. One disadvantage is that if bond prices fall a lot (below 8.5% or below), the covered call strategy may be more costly than just selling the put. Also, the covered call strategy can backfire if bond prices really take off. This is because Hanich loses wealth to the extent the bond price increases (through granting the purchaser of the call option the right to buy the bonds from Hanich at below market value). Given the volatility in either direction it might make more sense to buy a call and also sell a put. That way, Hanich can make money regardless of which way interest rates change. However, his main concern should be in protecting the fall in his bond portfolio which can decrease significantly if interest rates increase rapidly. 18. Determine the price of a European call option on a 6.5% four-year Treasury bond with a strike price of 100.25 and two years to expiration assuming: (1) the arbitrage-free binomial interest-rate tree shown in Exhibit 27-10 (based on a 10% volatility assumption), and (2) the price of the Treasury bond two years from now shown at each node. For each node, the value of the call option is the maximum of zero and the current price (P) minus the strike price (S), i.e., MAX(0; P – S). The current prices are given in Exhibit 27-11 as $97.9249, $100.4189, and $102.5335 for year two at three nodes which (from Chapter 17) we call HH, LH (or HL), and LL, respectively. At node HH, the value of the call option is: MAX(0; P – S) = MAX(0; $97.9249 – $100.25) = MAX(0; $2.3251) = $0. At node LH, the value of the call option is: MAX(0; P – S) = MAX(0; $100.4189 – $100.25) = MAX(0; $0.1689) = $0.1689. At node LL, the value of the call option is: MAX(0; P – S) = MAX(0; $102.5335 – $100.25) = MAX(0; $2.2835) = $2.2835. For year two we have two nodes that we will refer to as nodes H and L. At node H, the value is the average of the present value of the call option values at nodes HH and HL discounted by the corresponding rate from the binomial tree of 5.4289%. We have: call option value = [($0 / 1.054289) + ($0.1689 / 1.054289)] / 2 = [0 + 0.1602027] / 2 = 0.08010135 or about $0.0801. At node L, the value is the average of the present value of the call option values at nodes HL and LL discounted by the corresponding rate from the binomial tree of 4.4448%. We have: call option value = [($0.1689 / 1.04448) + ($2.2835 / 1.04448)] / 2 = [0.1617122 + 2.1863223] / 2 = 2.3480346 / 2 = 1.1740173 or about $1.1740. At the initial node (year zero), the value is the average of the present value of the call option values at nodes H and L discounted by the corresponding rate from the binomial tree of 3.500%. We have: call option value = [($0.08010135 / 1.035) + ($1.1740 / 1.035)] / 2 = [0.077392634 + 1.1342995] = 1.2116922 / 2 = 0.605846 or about $0.6058. Thus, the value of the call option is $0.60584 per $100 or about $6,058 per $1,000,000. 19. Determine the price of a European put option on a 6.5% four-year Treasury bond with a strike price of 100.25 and two years to expiration assuming the same information as in Exhibit 27-10. For each node the value of the put option is the maximum of zero and the strike price (S) minus the current price (P), i.e., MAX(0; S – P). The current prices are given in Exhibit 27-11 as $97.9249, $100.4189, and $101.5335 for year two at three nodes which (from Chapter 16) we call HH, LH (or HL), and LL, respectively. At node HH, the value of the put option is: MAX(0; S – P) = MAX(0; $100.25 – $97.9249) = MAX(0; $2.3251) = $2.3251. At node LH, the value of the put option is: MAX(0; S – P) = MAX(0; $100.25 – $100.4189) = MAX(0; $0.1689) = $0. At node LL, the value of the put option is: MAX(0; S – P) = MAX(0; $102.25 – $100.5335) = MAX(0; $2.2835) = $0. For year two we have two nodes that we will refer to as nodes H and L. At node H, the value is the average of the present value of the put option values at nodes HH and HL discounted by the corresponding rate from the binomial tree of 5.4289%. We have: put option value = [($2.3251 / 1.054289) + ($0 / 1.054289)] / 2 = [2.205373 + 0] / 2 = 1.102686 or about $1.1027. At node L, the value is the average of the present value of the put option values at nodes HL and LL discounted by the corresponding rate from the binomial tree of 4.4448%. We have: put option value = [($0 / 1.04448) + ($0 / 1.04448)] / 2 = [0 + 0] / 2 = 0 / 2 = $0. [NOTE. We already knew it was zero because both values were zero.] At the initial node (year zero), the value is the average of the present value of the put option values at nodes H and L discounted by the corresponding rate from the binomial tree of 3.500%. We have: put option value = [($1.102686 / 1.035) + ($0 / 1.035)] / 2 = [1.0653971 + 0] = 1.0653971 / 2 = 0.5326987 or about $0.5327. Thus, the value of the put option is $0.532699 per $100 or about $5,327 per $1,000,000. CHAPTER 28 INTEREST-RATE SWAPS, CAPS, AND FLOORS CHAPTER SUMMARY In Chapters 26 and 27, we discussed how interest-rate futures and options can be used to control interest-rate risk. There are other contracts useful for controlling such risk that commercial banks and investment banks can customize for their clients. These include (1) interest-rate swaps and options on swaps, and (2) interest-rate caps and floors and options on these agreements. In this chapter, we review each of them and explain how they can be used by institutional investors. INTEREST-RATE SWAPS In an interest-rate swap, two parties (called counterparties) agree to exchange periodic interest payments. The dollar amount of the interest payments exchanged is based on a predetermined dollar principal, which is called the notional principal amount. In the most common type of swap, one party agrees to pay the other party fixed-interest payments at designated dates for the life of the contract. This party is referred to as the fixed-rate payer or the floating-rate receiver. The other party, who agrees to make interest rate payments that float with some reference rate, is referred to as the floating-rate payer or fixed-rate receiver. Entering into a Swap and Counterparty Risk Interest-rate swaps are over-the-counter instruments. This means that they are not traded on an exchange. An institutional investor wishing to enter into a swap transaction can do so through either a securities firm or a commercial bank that transacts in swaps. The risks that parties take on when they enter into a swap are that the other party will fail to fulfill its obligations as set forth in the swap agreement. That is, each party faces default risk. The default risk in a swap agreement is called counterparty risk. Interpreting a Swap Position There are two ways that a swap position can be interpreted: (i) as a package of forward/futures contracts, and (ii) as a package of cash flows from buying and selling cash market instruments. Terminology, Conventions, and Market Quotes The date that the counterparties commit to the swap is called the trade date. The date that the swap begins accruing interest is called the effective date, and the date that the swap stops accruing interest is called the maturity date. The convention that has evolved for quoting swaps levels is that a swap dealer sets the floating rate equal to the index and then quotes the fixed-rate that will apply. The offer price that the dealer would quote the fixed-rate payer would be to pay 8.85% and receive LIBOR “flat” (“flat” meaning with no spread to LIBOR). The bid price that the dealer would quote the fixed-rate receiver would be to pay LIBOR flat and receive 8.75%. The bid-offer spread is 10 basis points. Calculation of the Swap Rate At the initiation of an interest-rate swap, the counterparties are agreeing to exchange future interest-rate payments and no upfront payments by either party are made. While the payments of the fixed-rate payer are known, the floating-rate payments are not known. This is because they depend on the value of the reference rate at the reset dates. For a LIBOR-based swap, the Eurodollar futures contract can be used to establish the forward (or future) rate for three-month LIBOR. In general, the floating-rate payment is determined as follows: floating-rate payment = notional amount × three-month LIBOR × . The equation for determining the dollar amount of the fixed-rate payment for the period is: fixed-rate payment = notional amount × swap rate × . It is the same equation as for determining the floating-rate payment except that the swap rate is used instead of the reference rate (three-month LIBOR in our illustration). Given the swap payments, we can demonstrate how to compute the swap rate. At the initiation of an interest-rate swap, the counterparties are agreeing to exchange future payments and no upfront payments by either party are made. This means that the swap terms must be such that the present value of the payments to be made by the counterparties must be at least equal to the present value of the payments that will be received. In fact, to eliminate arbitrage opportunities, the present value of the payments made by a party will be equal to the present value of the payments received by that same party. The equivalence (or no arbitrage) of the present value of the payments is the key principle in calculating the swap rate. We refer to the present value of $1 to be received in period t as the forward discount factor. In calculations involving swaps, we compute the forward discount factor for a period using the forward rates. These are the same forward rates that are used to compute the floating-rate payments—those obtained from the Eurodollar futures contract. We must make just one more adjustment. We must adjust the forward rates used in the formula for the number of days in the period (i.e., the quarter in our illustrations) in the same way that we made this adjustment to obtain the payments. Specifically, the forward rate for a period, which we will refer to as the period forward rate, is computed using the following equation: period forward rate = annual forward rate × . Given the payment for a period and the forward discount factor for the period, the present value of the payment can be computed. The forward discount factor is used to compute the present value of the both the fixed-rate payments and floating-rate payments. Beginning with the basic relationship for no arbitrage to exist: PV of floating-rate payments = PV of fixed-rate payments The formula for the swap rate is derived as follows. We begin with: fixed-rate payment for period t = notional amount × swap rate × . The present value of the fixed-rate payment for period t is found by multiplying the previous expression by the forward discount factor for period t. We have: present value of the fixed-rate payment for period t = notional amount × swap rate × × forward discount factor for period t. Summing up the present value of the fixed-rate payment for each period gives the present value of the fixed-rate payments. Letting N be the number of periods in the swap, we have: present value of the fixed-rate payments = swap rate × × × forward discount factor for period t. The condition for no arbitrage is that the present value of the fixed-rate payments as given by the expression above is equal to the present value of the floating-rate payments. We have: present value of floating-rate payments = swap rate × × × forward discount factor for period t. Solving for the swap rate gives swap rate = . The calculation of the swap rate for all swaps follows the same principle: equating the present value of the fixed-rate payments to that of the floating-rate payments. Valuing a Swap Once the swap transaction is completed, changes in market interest rates will change the payments of the floating-rate side of the swap. The value of an interest-rate swap is the difference between the present value of the payments of the two sides of the swap. Duration of a Swap As with any fixed-income contract, the value of a swap will change as interest rates change. Dollar duration is a measure of the interest-rate sensitivity of a fixed-income contract. From the perspective of the party who pays floating and receives fixed, the interest-rate swap position can be viewed as follows: long a fixed-rate bond + short a floating-rate bond. The implication here is that to increase the dollar duration of a portfolio, a manager should enter into a swap as the fixed-rate receiver. This is economically equivalent to leveraging the interest rate risk exposure of the portfolio and thereby adding dollar duration. By entering into a swap as the fixed-rate payer, instead, the manager reduces the dollar duration of the portfolio. Application to Portfolio Risk Control In Chapter 26, we explained how to use futures to alter the risk exposure of a portfolio to changes in interest rates (i.e., alter a portfolio’s duration). Earlier in this chapter we explained how an interest-rate swap is equivalent to a portfolio of forward/futures contracts. Therefore, it should be no surprise that interest-rate swaps can be used to provide the same duration-adjustment mechanism as futures contracts as illustrated in Chapter 26. Application of a Swap to Asset/Liability Management An interest-rate swap can be used to alter the cash flow characteristics of an institution’s assets so as to provide a better match between assets and liabilities. An interest-rate swap allows each party to accomplish its asset/liability objective of locking in a spread. An asset swap permits the two financial institutions to alter the cash flow characteristics of its assets: from fixed to floating or from floating to fixed. A liability swap permits two institutions to change the cash flow nature of their liabilities. Creation of Structured Notes Using Swaps Corporations can customize medium-term notes for institutional investors who want to make a market play on interest rate, currency, and/or stock market movements. That is, the coupon rate on the issue will be based on the movements of these financial variables. A corporation can do so in such a way that it can still synthetically fix the coupon rate. This can be accomplished by issuing an MTN and entering into a swap simultaneously. MTNs created in this way are called structured MTNs. Variants of the Generic Interest-Rate Swap Non-generic or individualized swaps have evolved as a result of the asset/liability needs of borrowers and lenders. These include swaps where the notional principal changes in a pre-determined way over the life of the swap and swaps in which both counterparties pay a floating rate. There are complex swap structures such as options on swaps (called swaptions) and swaps where the swap does not begin until some future time (called forward start swaps). What is important to appreciate is that these swap structures contain features that managers have found that they need to control interest-rate risk. Varying Notional Principal Amount Swaps In a generic or plain vanilla swap, the notional principal amount does not vary over the life of the swap. Thus it is sometimes referred to as a bullet swap. In contrast, for amortizing, accreting, and roller coaster swaps, the notional principal amount varies over the life of the swap. An amortizing swap is one in which the notional principal amount decreases in a predetermined way over the life of the swap. Such a swap would be used where the principal of the asset that is being hedged with the swap amortizes over time. Less common than the amortizing swap are the accreting swap and the roller coaster swap. An accreting swap is one in which the notional principal amount increases in a predetermined way over time. In a roller coaster swap, the notional principal amount can rise or fall from period to period. The terms of a generic interest-rate swap call for the exchange of fixed- and floating-rate payments. In a basis rate swap, both parties exchange floating-rate payments based on a different reference rate. The risk is that the spread between the prime rate and LIBOR will change. This is referred to as basis risk. Another popular swap is to have the floating leg tied to a longer-term rate such as the two-year Treasury note rather than a money market rate. Such a swap is called a constant maturity swap. A forward start swap is a swap wherein the swap does not begin until some future date that is specified in the swap agreement. A forward start swap will also specify the swap rate at which the counterparties agree to exchange payments commencing at the start date. Swaptions There are options on interest-rate swaps. These swap structures are called swaptions and grant the option buyer the right to enter into an interest-rate swap at a future date. The buyer of the swaption must pay the swaption seller a fee, the swaption price or premium. The time until expiration of the swap, the term of the swap, and the swap rate are specified. The swap rate is the strike rate for the swaption. A swaption can have either an American or European style exercise provision. There are two types of swaptions—a payer swaption and a receiver swaption. A payer swaption entitles the option buyer to enter into an interest-rate swap in which the option buyer pays a fixed rate and receives a floating rate. In a receiver swaption the swaption buyer has the right to enter into an interest-rate swap that requires paying a floating rate and receiving a fixed rate. INTEREST-RATE CAPS AND FLOORS An interest-rate agreement is an agreement between two parties whereby one party, for an upfront premium, agrees to compensate the other at specific time periods if a designated interest rate, called the reference rate, is different from a predetermined level. When one party agrees to pay the other when the reference rate exceeds a predetermined level, the agreement is referred to as an interest-rate cap or ceiling. The agreement is referred to as an interest-rate floor when one party agrees to pay the other when the reference rate falls below a predetermined level. The predetermined interest-rate level is called the strike rate. Interest-rate caps and floors can be combined to create an interest-rate collar. This is done by buying an interest-rate cap and selling an interest-rate floor. Some commercial banks and investment banking firms write options on interest-rate agreements for customers. Options on caps are captions; options on floors are called flotions. Risk/Return Characteristics In an interest-rate agreement, the buyer pays an upfront fee representing the maximum amount that the buyer can lose and the maximum amount that the writer of the agreement can gain. The only party that is required to perform is the writer of the interest-rate agreement. The buyer of an interest-rate cap benefits if the underlying interest rate rises above the strike rate because the seller (writer) must compensate the buyer. The buyer of an interest rate floor benefits if the interest rate falls below the strike rate, because the seller (writer) must compensate the buyer. Valuing Caps and Floors The arbitrage-free binomial model can be used to value a cap and a floor. This is because a cap and a floor are nothing more than a package or strip of options. More specifically, they are a strip of European options on interest rates. Thus to value a cap the value of each period’s cap, called a caplet, is found and all the caplets are then summed. The same can be done for a floor. Similarly, an interest rate floor can be valued. The value for the floor for any year is called a floorlet. Applications To see how interest-rate agreements can be used for asset/liability management, consider the problems faced by a commercial bank which needs to lock in an interest-rate spread over its cost of funds. Because it borrows short term, its cost of funds is uncertain. The bank may be able to purchase a cap, however, so that the cap rate plus the cost of purchasing the cap is less than the rate it is earning on its fixed-rate commercial loans. If short-term rates decline, the bank does not benefit from the cap, but its cost of funds declines. The cap therefore allows the bank to impose a ceiling on its cost of funds while retaining the opportunity to benefit from a decline in rates. The bank can reduce the cost of purchasing the cap by selling a floor. In this case the bank agrees to pay the buyer of the floor if the reference rate falls below the strike rate. The bank receives a fee for selling the floor, but it has sold off its opportunity to benefit from a decline in rates below the strike rate. By buying a cap and selling a floor the bank creates a “collar” with a predetermined range for its cost of funds. KEY POINTS An interest-rate swap is an agreement specifying that the parties exchange interest payments at designated times. In a generic interest-rate swap, one party will make fixed-rate payments (called the fixed-rate payer), and the other will make floating-rate payments (called the fixed-rate receiver), with payments based on the notional principal amount. Asset and risk managers use interest-rate swaps to alter the duration of a portfolio, alter the cash flow characteristics of their assets or liabilities, or to capitalize on perceived capital market inefficiencies. A swap position can be interpreted as either a package of forward/futures contracts or a package of cash flows from buying and selling cash market instruments. The swap rate is computed by finding the rate that will make the present value of the cash flow of both sides of the swap equal. The value of an existing swap is equal to the difference in the present value of the two payments. The interest-rate sensitivity or duration of a swap from the perspective of a floating-rate payer is just the difference between the duration of the fixed-rate bond and duration of the floating-rate bond that compose the swap. Most of the interest-rate sensitivity of a swap will result from the duration of the fixed-rate bond since the duration of the floating-rate bond will be small. Non-generic swaps include swaps where the notional amount changes in a predetermined way over the life of the swap (amortizing, accreting, and roller coaster swaps) and swaps in which both counterparties pay a floating rate (basis swaps and Constant Maturity Treasury swaps). There are complex swap structures, such as swaps where the swap does not begin until some future time (forward start swaps) and options on swaps (swaptions). Swaptions can be used to create a portfolio with option-type payoffs that will have the desired duration if rates move in a favorable direction but limit adverse movements when interest rates moves in the opposite direction. The cost of creating such a favorable risk-return relationship is the cost of the swaptions. An interest-rate agreement allows one party for an upfront premium the right to receive compensation from the writer of the agreement if a designated interest rate is different from a predetermined level. An interest-rate cap calls for one party to receive a payment if a designated interest rate is above the strike rate. An interest-rate floor lets one party receive a payment if a designated interest rate is below the strike rate. An interest-rate cap can be used to establish a ceiling on the cost of funding; an interest-rate floor can be used to establish a floor return. Buying a cap and selling a floor creates a collar. A cap and a floor can be valued using the binomial model. ANSWERS TO QUESTIONS FOR CHAPTER 28 (Questions are in bold print followed by answers.) 1. Suppose that a dealer quotes these terms on a five-year swap: fixed-rate payer to pay 4.4% for LIBOR and fixed-rate receiver to pay LIBOR for 4.2%. Answer the below questions. (a) What is the dealer’s bid-asked spread? Dealer’s bid-asked spread = (offer price dealer quotes fixed-rate payer) – (bid price dealer quotes fixed-rate receiver)  Dealer’s bid-offer spread = 4.40% – 4.20% = 0.20% or 0.0020 or 20 basis points. (b) How would the dealer quote the terms by reference to the yield on five-year Treasury notes? The fixed rate is some spread above the Treasury yield curve with the same term to maturity as the swap. Suppose the five-year Treasury yield is 9.0%. Then the offer price that the dealer would quote to the fixed-rate payer is the five-year Treasury rate plus 50 basis points versus receiving LIBOR flat. For the fixed-rate receiver, the bid price quoted would be LIBOR flat versus the five-year Treasury rate plus 20 basis points. The dealer would quote such a swap as 20-50, meaning that the dealer is willing to enter into a swap to receive LIBOR and pay a fixed rate equal to the five-year Treasury rate plus 20 basis points; and it would be willing to enter into a swap to pay LIBOR and receive a fixed rate equal to the five-year Treasury rate plus 50 basis points. The difference between the Treasury rate paid and received is the bid-offer spread. 2. Give two interpretations of an interest-rate swap. There are two ways that a swap position can be interpreted: (i) as a package of forward/futures contracts, and (ii) as a package of cash flows from buying and selling cash market instruments. 3. In determining the cash flow for the floating-rate side of a LIBOR swap, explain how the cash flow is determined. Assume a swap of 12 quarterly floating-rate payments for three years with the first quarter consisting of 90 days from January 1st of year 1 to March 31st of year 1 assuming a non-leap year. The cash flow for this period is: floating-rate payment = notional amount × three-month LIBOR × . Note that each futures contract is for $1 million and hence 100 contracts have a notional amount of $100 million. Let’s assume $100 million notional amount and a LIBOR of 5%. The cash flow for period 1 is: payment = $100,000,000 × 0.05 × 0.25 = $1,250,000. While this first quarterly payment is known, the next 11 are not. The second quarterly payment, from April 1 of year 1 to June 30 of year 1, has 91 days. The floating-rate payment is determined by three-month LIBOR on April 1 of year 1 and paid on June 30 of year 1. This is achieved by looking at the three-month Eurodollar futures contract for settlement on June 30 of year 1. That futures contract provides the rate that can be locked in for three-month LIBOR on April 1 of year 1. We refer to that rate for three-month LIBOR as the forward rate. Therefore, if the fixed-rate payer bought 100 contract of these three-month Eurodollar futures contracts on January 1 of year 1 (the inception of the swap) that settle on June 30 of year 1, then the payment that will be locked in for the second quarter (April 1 to June 30 of year 1) is payment = notional amount × annual forward rate × . Given that the notional amount is $100 million and the number of days is 91, let us assume the annual forward rate is 5.2%. Using these numbers, the payment is: fixed-rate payment = $100,000,000 × 0.052 × = $1,314,444.44. Similarly, the Eurodollar futures contract can be used to lock in a floating-rate payment for each of the next 10 quarters. It is important to emphasize that the reference rate at the beginning of period t determines the floating rate that will be paid for the period. However, the floating-rate payment is not made until the end of period t. 4. How is the swap rate calculated? To compute the swap rate, we begin with the basic relationship for no arbitrage to exist: present value of fixed-rate payments = present value of floating-rate payments. For the fixed-rate payment for period t, we have: fixed-rate payment = notional amount × swap rate × The present value of the fixed-rate payment for period t is found by multiplying the fixed-rate payment expression by the forward discount factor for period t. That is, we have: present value of the fixed-rate payment for period t = notional amount × swap rate × × forward discount factor for period t. Summing up the present value of the fixed-rate payment for each period gives the present value of the fixed-rate payments. Letting N be the number of periods in the swap, we have: present value of the fixed-rate payments = swap rate × × × forward discount factor for period t. The condition for no arbitrage is that the present value of the fixed-rate payments as given by the expression above is equal to the present value of the floating-rate payments. That is, we have: present value of floating-rate payments = swap rate × × × forward discount factor for period t. Solving for the swap rate, we have: swap rate = . 5. Suppose that a life insurance company has issued a three-year GIC with a fixed-rate of 10%. Under what circumstances might it be feasible for the life insurance company to invest the funds in a floating-rate security and enter into a three-year interest-rate swap in which it pays a floating rate and receives a fixed-rate? If the life insurance can enter a swap that guarantees a satisfactory spread above the 10% it is committed to pay, then it would not only be feasible but desirable to enter into the swap. More details are given below. Suppose the life insurance company can enter into a swap with a bank which has a portfolio consisting of three-year term commercial loans with a fixed interest rate. The principal value of bank’s portfolio is $10 million, and the interest rate on all its loans in its portfolio is 11%. The loans are interest-only loans; interest is paid semiannually, and the principal is paid at the end of three years. That is, assuming no default on the loans, the cash flow from the loan portfolio is $1.1 million every six months for the next three years and $10 million at the end of three years (in addition to the $1.1 million interest). To fund its loan portfolio, assume that the bank is relying on the issuance of six-month certificates of deposit. The interest rate that the bank plans to pay on its six-month CDs is six-month LIBOR plus 40 basis points. The risk that the bank faces is that six-month LIBOR will be 10.6% or greater. To understand why, remember that the bank is earning 11% annually on its commercial loan portfolio. If six-month LIBOR is 10.6%, it will have to pay 10.6% plus 40 basis points, or 11%, to depositors for six-month funds and there will be no spread income. Worse, if six-month LIBOR rises above 10.6%, there will be a loss; that is, the cost of funds will exceed the interest rate earned on the loan portfolio. The bank’s objective is to lock in a spread over the cost of its funds. The life insurance company can seize this opportunity to cover its commitment to pay a 10% rate for the next three years on the $10 million GIC it has issued. The amount of its GIC is $10 million. Suppose that the life insurance company has the opportunity to invest $10 million in what it considers an attractive three-year floating-rate instrument in a private placement transaction. The interest rate on this instrument is six-month LIBOR plus 160 basis points. The coupon rate is set every six months. The risk that the life insurance company faces in this instance is that six-month LIBOR will fall so that the company will not earn enough to realize a spread over the 10% rate that it has guaranteed to the GIC holders. If six-month LIBOR falls to 8.4% or less, no spread income will be generated. To understand why, suppose that six-month LIBOR at the date the floating-rate instrument resets its coupon is 8.4%. Then the coupon rate for the next six months will be 10% (8.4% plus 160 basis points). Because the life insurance company has agreed to pay 10% on the GIC policy, there will be no spread income. Should six-month LIBOR fall below 8.4%, there will be a loss. We can summarize the asset/liability problems of the bank and the life insurance company as follows. Bank: (i) Has lent long term and borrowed short term. (ii) If six-month LIBOR rises, spread income declines. Life Insurance Company: (i) Has lent short term and borrowed long term. (ii) If six-month LIBOR falls, spread income declines. Now let’s suppose the market has available a three-year interest-rate swap with a notional principal amount of $10 million. The swap terms available to the bank are as follows: (i) Every six months the bank will pay 9.45% (annual rate). (ii) Every six months the bank will receive LIBOR. The swap terms available to the insurance company are as follows: (i) Every six months the life insurance company will pay LIBOR. (ii) Every six months the life insurance company will receive 9.40%. What has this interest-rate contract done for the bank and the life insurance company? Consider first the bank. For every six-month period for the life of the swap agreement, the interest-rate spread will be as follows: Annual Interest Rate Received: From commercial loan portfolio 11.00% From interest-rate swap Six-month LIBOR Total 11.00% + six-month LIBOR Annual Interest Rate Paid: To CD depositors six-month LIBOR On interest-rate swap 9.45% Total 9.45% + six-month LIBOR Outcome: To be received 11.00% + six-month LIBOR To be paid 9.45% + six-month LIBOR Spread income 1.55% or 155 basis points Thus, whatever happens to six-month LIBOR, the bank locks in a spread of 155 basis points. Now let’s look at the effect of the interest-rate swap from the perspective of the life insurance company: Annual Interest Rate Received: From floating-rate instrument 1.6% + six-month LIBOR From interest-rate swap 9.40% Total 11.00% + six-month LIBOR Annual Interest Rate Paid: To GIC policyholders 10.00% On interest-rate swap Six-month LIBOR Total 10.00% + six-month LIBOR Outcome: To be received 11.00% + six-month LIBOR To be paid 10.00% + six-month LIBOR Spread income 1.00% or 100 basis points Thus, whatever happens to six-month LIBOR, the insurance company locks in a spread of 100 basis points. The interest-rate swap has allowed each party to accomplish its asset/liability objective of locking in a spread. It permits the two financial institutions to alter the cash flow characteristics of its assets: from fixed to floating in the case of the bank, and from floating to fixed in the case of the life insurance company. 6. How can interest rate swap be used to reduce the duration of portfolio to match the duration of a benchmark? To reduce the duration so as to match the benchmark, the manager can enter into a swap as the fixed-rate payer. If the manager wanted to increase the duration, a position in a swap can be taken to be a fixed-rate receiver. More details are given below. As with any fixed-income contract, the value of a swap will change as interest rates change. Dollar duration is a measure of the interest-rate sensitivity of a fixed-income contract. From the perspective of the party who pays floating and receives fixed, the interest-rate swap position can be viewed as follows: long a fixed-rate bond + short a floating-rate bond. This means that the dollar duration of an interest-rate swap from the perspective of a fixed-rate receiver is simply the difference between the dollar duration of the two bond positions that make up the swap; that is, dollar duration of a swap = dollar duration of a fixed-rate bond – dollar duration of a floating-rate bond Most of the dollar price sensitivity of a swap due to interest-rate changes will result from the dollar duration of the fixed-rate bond because the dollar duration of the floating-rate bond will be small. The closer the swap is to the date that the coupon rate is reset, the smaller the dollar duration of a floating-rate bond. The implication here is that to increase the dollar duration of a portfolio, a manager should enter into a swap as the fixed-rate receiver. This is economically equivalent to leveraging the interest rate risk exposure of the portfolio and thereby adding dollar duration. By entering into a swap as the fixed-rate payer, instead, the manager reduces the dollar duration of the portfolio. We now illustrate how an interest-rate swap can reduce the duration to match the duration of a benchmark. Suppose the manager of a portfolio with a market value as of December 31, 2012 of $48,109,810 has a benchmark that is the Barclays Capital Intermediate Aggregate Index. On December 31, 2012, the duration of the index and the portfolio were 2.97 and 3.68, respectively. The manager wants to restructure the portfolio so that the portfolio’s duration matches that of the benchmark. That is, the portfolio manager seeks to follow a duration-neutral strategy and therefore the portfolio’s target duration is 2.97. We know that for a 100 basis point change in rates: portfolio target dollar duration = 2.97% × $48,109,810 = $1,428,594 portfolio current dollar duration = 3.68% × $48,109,810 = $1,770,110 The difference between the target and the current dollar duration for the portfolio is $341,516. This means that to get to the target portfolio duration of 2.97, the portfolio manager must decrease the dollar duration of the current portfolio by $341,516 for a 100 basis change in interest rates. On December 31, 2012, the manager decides to use a 5-year interest-rate swap at-the-money (i.e., par) swap. The swap rate is 2.42%. Since the objective is to decrease the dollar duration, then this involves the manager entering into a swap as the fixed-rate payer. It can be shown that the dollar duration for a notional amount of $7,699,779 is $340,881. Hence a 5-year interest-rate swap with that notional amount combined with the current portfolio will result in a new dollar duration for the portfolio of $1,769,475, which is close to the target dollar duration needed to obtain a portfolio duration of 2.97. 7. A portfolio manager buys a swaption with a strike rate of 4.5% that entitles the portfolio manager to enter into an interest-rate swap to pay a fixed-rate and receives a floating rate. The term of the swaption is five years. Answer the below questions. (a) Is this swaption a payer swaption or a receiver swaption? Explain. It is a payer swaption because it entitles the option buyer to enter into an interest-rate swap in which the buyer of the option pays a fixed-rate and receives a floating rate. More details are given below. There are options on interest-rate swaps. These swap structures are called swaptions and grant the option buyer the right to enter into an interest-rate swap at a future date. The time until expiration of the swap, the term of the swap, and the swap rate are specified. The swap rate is the strike rate for the swaption. The swaption has the European-type exercise provision. That is, the option can be exercised only at the option’s expiration date. There are two types of swaptions—a payer swaption and a receiver swaption. A payer swaption entitles the option buyer to enter into an interest-rate swap in which the buyer of the option pays a fixed-rate and receives a floating rate. For example, suppose that the strike rate is 7.25%, the term of the swap is five years, and the swaption expires in two years. Also assume that it is an American-type exercise provision. This means that the buyer of this swaption has the right within the next two year to enter into a three-year interest-rate swap in which the buyer pays 7.25% (the swap rate, which is equal to the strike rate) and receives the reference rate. In a receiver swaption the buyer of the swaption has the right to enter into an interest-rate swap that requires paying a floating rate and receiving a fixed-rate. For example, if the strike rate is 5.75%, the swap term is five years, and the option expires in one year, the buyer of this receiver swaption has the right within the next year until the option expires (assuming it is an American-type exercise provision) to enter into a four-year interest-rate swap in which the buyer receives a swap rate of 5.75% (i.e., the strike rate) and pays the reference rate. (b) What does the strike rate of 4.5% mean? A strike rate of 4.5% means that the buyer of this swaption has the right to enter into a five-year interest-rate swap in which the buyer pays 4.5% (the swap rate, which is equal to the strike rate) and receives the reference rate. 8. The following appeared on a quote sheet: “Receiver Swaption: An option to receive the fixed leg of a swap (i.e., long receiver is long duration). Payer Swaption: An option to pay the fixed leg of a swap (i.e., long payer is short duration)”. (a) Explain why for the receiver swaption the party is long duration. In a receiver swaption the swaption buyer has the right to enter into an interest-rate swap that requires paying a floating rate and receiving a fixed rate. As with any fixed-income contract, the value of a swap will change as interest rates change. Dollar duration is a measure of the interest-rate sensitivity of a fixed-income contract. From the perspective of the party who pays floating and receives fixed, the interest-rate swap position can be viewed as follows: long a fixed-rate bond + short a floating-rate bond. This means that the dollar duration of an interest-rate swap from the perspective of a fixed-rate receiver is simply the difference between the dollar duration of the two bond positions that make up the swap; that is, dollar duration of a swap = dollar duration of a fixed-rate bond – dollar duration of a floating-rate bond Most of the dollar price sensitivity of a swap due to interest-rate changes will result from the dollar duration of the fixed-rate bond because the dollar duration of the floating-rate bond will be small. The closer the swap is to the date that the coupon rate is reset, the smaller the dollar duration of a floating-rate bond. The implication here is that to increase the dollar duration of a portfolio, a manager should enter into a swap as the fixed-rate receiver. This is economically equivalent to leveraging the interest rate risk exposure of the portfolio and thereby adding dollar duration. By entering into a swap as the fixed-rate payer, instead, the manager reduces the dollar duration of the portfolio. More details are given below. If we want to add duration to a portfolio and a fixed-rate receiver swap will add duration, this means that the manager should buy a receiver swaption (i.e., receive fixed and pay floating). Suppose on March 31, 2011, the manager decides to use a 1×5 ATM receiver European swaption with a strike rate (swap rate) of 3.26%. Assuming a 115 basis point volatility, the cost of a receiver swaption that would be needed to produce the desired interest rate exposure is $307,705 (the notional amount of the receiver swaption is $14,487,071). The dollar duration that would be added to the portfolio using the receiver swaption would be $340,881 (the same as in the fixed-rate swap). As we explained in describing options in Chapter 27, an option can be in-the-money (ITM), at-the-money (ATM), or out-of-the-money (OTM). The terminology applies based on how the strike price differs from the prevailing market price of the underlying. The same holds for swaptions. Typically dealers quote ATM swaptions based on an assumed volatility. ITM and OTM swaptions when quotes are based on higher volatilities than ATM options. How is a swaption used? Its usefulness can be found in two applications of a swap: controlling the duration of a portfolio and asset-liability management. Below we focus on the duration aspect. In Chapter 27 we illustrated how to use futures options to control the interest rate risk of an individual bond. Let’s look at how a swaption can be used to create an option-type payoff for the bond portfolio that was used in Chapter 26 to show to use futures to change duration and earlier in this chapter to show how to use swaps for the same purpose. Let’s look at the type of position that must be taken. Recall in the illustration that the manager wants to increase the interest rate risk exposure of the portfolio. With a futures contract this is done by buying Treasury futures and with an interest rate swap this accomplished by being the fixed-rate receiver. Using an option-type instrument such as a swaption, the manager wants a non-linear payoff whereby if interest rates decline, the portfolio will have a payoff similar to the benchmark that has a duration of 3.68. Yet, if interest rates rise, the portfolio does not decline in value by as much as a portfolio with a duration of 3.68. Increasing upside potential but limiting downside risk is what characterizes option-type instruments such as a swaption. Of course, this is not free; it requires the payment of a fee, the swaption premium. What position in the swaption should the manager take? Since the manager wants to add duration to the portfolio and a fixed-rate receiver swap will add duration, this means that the manager would buy a receiver swaption (i.e., receive fixed and pay floating). Suppose on March 31, 2011, the manager decides to use a 1×5 ATM receiver European swaption with a strike rate (swap rate) of 3.26%. Assuming a 115 basis point volatility, the cost of a receiver swaption that would be needed to produce the desired interest rate exposure is $307,705 (the notional amount of the receiver swaption is $14,487,071). The dollar duration that would be added to the portfolio using the receiver swaption would be $340,881 (the same as in the fixed-rate swap). (b) Explain why for the payer swaption the party is short duration. A payer swaption entitles the option buyer to enter into an interest-rate swap in which the option buyer pays a fixed rate and receives a floating rate. As noted in the previous question: dollar duration of a swap = dollar duration of a fixed-rate bond – dollar duration of a floating-rate bond Most of the dollar price sensitivity of a swap due to interest-rate changes will result from the dollar duration of the fixed-rate bond because the dollar duration of the floating-rate bond will be small. The closer the swap is to the date that the coupon rate is reset, the smaller the dollar duration of a floating-rate bond. The implication here is that to decrease the dollar duration of a portfolio, a manager should enter into a swap as the fixed-rate payer. This is economically equivalent to un-levering the interest rate risk exposure of the portfolio and thereby reducing dollar duration. 9. The manager of a savings and loan association is considering the use of a swap as part of its asset/liability strategy. The swap would be used to convert the payments of its portfolio of fixed-rate residential mortgage loans into a floating payment. Answer the below questions. (a) What is the risk with using a plain vanilla or generic interest-rate swap? Assuming its liabilities are composed of floating-rate payments, converting its assets (i.e., its portfolio of fixed-rate mortgage loans) in floating payment should be risk reducing. However, the risk reduction depends on the reference rates used as well as the extent the liabilities are matched with the assets. Relatedly, there is the risk that the floating rate will not be as great as the fixed-rate currently being received. The S&L would want to make sure it obtains the spread necessary to make a profit. In terms of the generic interest-rate swap itself, there is counterparty risk or risk that the other party will default on its payments. Although a default by one party would not mean any principal was lost because the notional principal amount had not been exchanged, it would mean that the objective for which the swap was entered into would be impaired. (b) Why might a manager consider using an interest-rate swap in which the notional principal amount declines over time? A manager would consider using an interest-rate swap in which the notional principal amount declines over time where the principal of the asset that is being hedged with the swap amortizes over time. Such a swap is called an amortizing swap. More details are given below. An amortizing swap is one in which the notional principal amount decreases in a predetermined way over the life of the swap. Such a swap would be used where the principal of the asset that is being hedged with the swap amortizes over time. For example, consider an asset/liability problem faced by the bank where its commercial loans are assumed to pay interest every six months and repay principal only at the end of the loan term. However, what if the commercial loan is a typical term loan; that is, suppose it is a loan that amortizes. In such circumstances, the outstanding principal for the loans would decline and the bank would need a swap where the notional principal amount amortizes in the same way as the loans. (c) Why might a manager consider buying a swaption? Managers will consider buying a swaption if at a later date they plan on entering into a swap and need to lock in a desired swap rate that they will either pay or receive. More details on swaptions are given below. There are options on interest-rate swaps. These swap structures are called swaptions and grant the option buyer the right to enter into an interest-rate swap at a future date. The time until expiration of the swap, the term of the swap, and the swap rate are specified. The swap rate is the strike rate for the swaption. The swaption has the American-type exercise provision. That is, the option can be exercised only at the option’s expiration date. There are two types of swaptions—a payer swaption and a receiver swaption. A payer swaption entitles the option buyer to enter into an interest-rate swap in which the buyer of the option pays a fixed-rate and receives a floating rate. For example, suppose that the strike rate is 7%, the term of the swap is three years, and the swaption expires in two years. Also assume that it is an American-type exercise provision. This means that the buyer of this swaption at the end of two years has the right to enter into a three-year interest-rate swap in which the buyer pays 7% (the swap rate, which is equal to the strike rate) and receives the reference rate. In a receiver swaption the buyer of the swaption has the right to enter into an interest-rate swap that requires paying a floating rate and receiving a fixed-rate. For example, if the strike rate is 6.25%, the swap term is five years, and the option expires in one year, the buyer of this receiver swaption has the right when the option expires in one year (assuming it is an American-type exercise provision) to enter into a four-year interest-rate swap in which the buyer receives a swap rate of 6.25% (i.e., the strike rate) and pays the reference rate. 10. Consider the following interest-rate swap: • the swap starts today, January 1 of year 1 (swap settlement date) • the floating-rate payments are made quarterly based on actual / 360 • the reference rate is three-month LIBOR • the notional amount of the swap is $40 million • the term of the swap is three years Answer the below questions. (a) Suppose that today’s three-month LIBOR is 5.7%. What will the fixed-rate payer for this interest rate swap receive on March 31 of year 1 (assuming that year 1 is not a leap year)? The quarterly floating-rate payments are based on an actual / 360 day count convention. This convention means that 360 days are assumed in a year and that in computing the interest for the quarter the actual number of days in the quarter is used. The floating-rate payment is set at the beginning of the quarter but paid at the end of the quarter—that is, the floating-rate payments are made in arrears. For our problem, today’s three-month LIBOR is 5.7%. Thus, the fixed-rate payer receives payment based on this rate on March 31 of year 1—the date when the first quarterly swap payment is made. There is no uncertainty about what this floating-rate payment will be. In general, the floating-rate payment is given as: floating-rate payment = notional amount × three-month LIBOR × . In our problem, assuming a non-leap year, the number of days from January 1 of year 1 to March 31 of year 1 (the first quarter) is 90. If three-month LIBOR is 5.7%, then the fixed-rate payer will receive a floating-rate payment on March 31 of year 1 as shown below: floating-rate payment = $40,000,000 × 0.057 × = $570,000. (b) Assume the Eurodollar futures price for the next seven quarters is as follows: Quarter Starts Quarter Ends Number of Days in Quarter Eurodollar Futures Price April 1 year 1 June 30 year 1 91 94.10 July 1 year 1 Sept 30 year 1 92 94.00 Oct 1 year 1 Dec 31 year 1 92 93.70 Jan 1 year 2 Mar 31 year 2 90 93.60 April 1 year 2 June 30 year 2 91 93.50 July 1 year 2 Sept 30 year 2 92 93.20 Oct 1 year 2 Dec 31 year 2 92 93.00 Compute the forward rate for each quarter. Forward rate for period 1 is: three-month LIBOR today × . Inserting in our values, we have: 5.70% × = 1.425%. The forward rate for periods 2 through 8 is given by: forward rate = × Inserting in the value for period two (i.e., April 1 year1 to June 30 year 1), we have: forward rate = × = 0.0149138 or 1.4913889%. Similarly, for periods 3, 4, 5, 6, 7, and 8, the respective forward rates are: 1.5333333%, 1.6100%, 1.6000%, 1.6430556%, 1.7377778%, and 1.7888889%. (c) What is the floating-rate payment at the end of each quarter for this interest-rate swap? The floating-rate payment for each period is the forward rate for that period, as given in the part (b), times the notional amount of $40 million. For period one, we have forward rate for period 1 × notional amount = 0.01425 × $40,000,000 = $570,000.00. For period two, we have: forward rate for period 2 × notional amount = 0.014913889 × $40,000,000 = $596,555.56. Similarly, for periods 3, 4, 5, 6, 7, and 8, the respective forward rates are: $613,333.33, $644,000.00, $640,000.00, $657,222.22, $695,111.11, and $715,555.56. 11. Answer the below questions. (a) Assume that the swap rate for an interest-rate swap is 7% and that the fixed-rate swap payments are made quarterly on an actual / 360 basis. If the notional amount of a two-year swap is $20 million, what is the fixed-rate payment at the end of each quarter assuming the following number of days in each quarter? Period Quarter Days in Quarter 1 92 2 92 3 90 4 91 5 92 6 92 7 90 8 91 The fixed-rate payment for each quarter is given by: fixed-rate payment = notional amount × swap rate × where the notional amount is $20 million, the swap rate is 7%, and the number of days is the number for that period. For period one, we have: fixed-rate payment = notional amount × swap rate × . Inserting in our values, we have: fixed-rate payment for period 1 = 0.07 × $20,000,000 × = $357,777.78. Similarly, for periods 2, 3, 4, 5, 6, 7, and 8, the respective fixed-rate payments are: $357,777.78, $350,000.00, $353,888.89, $357,777.78, $357,777.78, $350,000.00, and $353,888.89. (b) Assume that the swap in part (a) requires payments semiannually rather than quarterly. What is the semiannual fixed-rate payment? First, we need the days for each of the four semiannual periods for the two years. Period one’s days are 92 + 92 = 184. Period two’s days are: 90 + 91 = 181. Period three’s days are: 92 + 92 = 184. Period four’s days are 90 + 91 = 181. We use the formula given above as: fixed-rate payment = notional amount × swap rate × . where the notional amount and swap rate are the same but the number of days change as given above. Inserting in our values, we get for the first period: fixed-rate payment for period 1 = 0.07 × $20,000,000 × = $715,555.56. Similarly, for periods 2, 3, and 4, the respective fixed-rate payments are: $703,888.89, $715,555.56, and $703,888.89. (c) Suppose that the notional amount for the two-year swap is not the same in both years. Suppose instead that in year 1 the notional amount is $20 million, but in year 2 the notional amount is $12 million. What is the fixed-rate payment every six months? The fixed-payments for the first two six-month periods are the same as given in part (b) as $715,555.56 and $703,888.89. For period three, the fixed-rate payment is: fixed-rate payment for period 3 = 0.07 × $12,000,000 × = $429,333.33. Similarly, for period four, the fixed-rate payment is: fixed-rate payment for period 4 = 0.07 × $12,000,000 × = $422,333.33. 12. Given the current three-month LIBOR and the Eurodollar futures prices shown in the table below, compute the forward rate and the forward discount factor for each period. Period Days in Quarter 3-month LIBOR Current Eurodollar Futures Price 1 90 5.90% 2 91 93.90 3 92 93.70 4 92 93.45 5 90 93.20 6 91 93.15 For period one, we haves: forward rate for period 1 = three-month LIBOR today × . Inserting in our values, we get: 5.90% × = 1.475%. The forward rate for periods 2 through 6 is given by: forward rate = × . Inserting in the value for period two, we have: forward rate for period 2 = × = 0.015419444 or 1.5419444%. Similarly, for periods 3, 4, 5, and 6, the respective forward rates are: 1.6100%, 1.6738889%, 1.7000%, and 1.7315278%. The forward discount factor for period t is given by: 1 / [(1 + forward rate period 1)(1 + forward rate period 2) . . . (1 + forward rate period t)]. For period 1, the forward discount factor is: forward discount factor = 1 / (1 + forward rate for period 1) = 1 / 1.01475 = 0.98546440. For period 2, the forward discount factor is: forward discount factor = 1 / [(1 + forward rate period 1)(1 + forward rate period 2)] = 1 / [(1.01475)(1.015419444)] = 0.97049983. Similarly for periods 3, 4, 5, and 6, the respective forward discount factors are: 0.95512236, 0.93939789, 0.92369507, and 0.90797326. 13. Answer the below questions. (a) Suppose that at the inception of a five-year interest-rate swap in which the reference rate is three-month LIBOR the present value of the floating-rate payments is $16,555,000. The fixed-rate payments are assumed to be semiannual. Assume also that the following is computed for the fixed-rate payments (using the notation in the chapter): × × forward discount factor for period t = $236,500,000. What is the swap rate for this swap? The swap rate is given by: swap rate = . Inserting in our values we get: swap rate = = 0.0700 or 7.00%. (b) Suppose that the five-year yield from the on-the-run Treasury yield curve is 6.4%. What is the swap spread? Given the swap rate, the swap spread can be determined. For example, since this is a five-year swap, the convention is to use the five-year on-the-run Treasury rate as the benchmark. Because the yield on that issue is 6.40%, the swap spread is 7.00% – 6.40% = 0.6% or 60 basis points. 14. An interest-rate swap had an original maturity of five-years. Today, the swap has two years to maturity. The present value of the fixed-rate payments for the remainder of the term of the swap is $910,000. The present value of the floating-rate payments for the remainder of the swap is $710,000. Answer the below questions. (a) What is the value of this swap from the perspective of the fixed-rate payer? We have: present value of fixed-rate payments = $910,000 and present value of floating-rate payments = $710,000. The two present values are not equal, therefore, for one party the value of the swap increased while for the other party the value of the swap decreased. The fixed-rate payer will receive the floating-rate payments. These floating-rate payments have a present value of $710,000. The present value of the payments that must be made by the fixed-rate payer is $910,000. Thus, the swap has a negative value for the fixed-rate payer equal to the difference in the two present values of $710,000 – $910,000 = $200,000.00. (b) What is the value of this swap from the perspective of the fixed-rate receiver? The fixed-rate receiver will receive the fixed-rate payments. These fixed-rate payments have a present value of $910,000. The present value of the payments that must be made by the fixed-rate payer is $710,000. Thus, the swap has a positive value for the fixed-rate receiver equal to the difference in the two present values of $910,000 – $710,000 = $200,000.00. 15. Suppose that a savings and loan association buys an interest-rate cap that has these terms: The reference rate is the six-month Treasury bill rate; the cap will last for five years; payment is semiannual; the strike rate is 5.5%; and the notional amount is $10 million. Suppose further that at the end of a six-month period, the six-month Treasury bill rate is 6.1%. Answer the below questions. (a) What is the amount of the payment that the savings and loan association will receive? The buyer of a cap benefits if the interest rate rises above the strike rate. Under this agreement, every six months for the next five years, the savings and loan association (S&L) will receive whenever six-month LIBOR exceeds the strike price of 5.5%. The payment will equal the dollar value of the difference between six-month LIBOR and 5.5% times the notional principal amount divided by 2. For our problem, if six months from now six-month LIBOR is 6.1%, the S&L will receive: (0.061 – 0.055)($10,000,000) / 2 = $30,000. (b) What would the writer of this cap pay if the six-month Treasury rate were 5.45% instead of 6.1%? If six-month LIBOR is 5.5% or less, the writer would not pay anything to the S&L. Since 5.45% is less than 5.5%, the writer pays nothing. 16. What is the relationship between an interest-rate agreement and an option on an interest rate? An interest-rate agreement is a package of interest-rate options. More details are given below. Just like a standard option contract, the buyer of an interest-rate agreement pays an upfront fee which represents the maximum amount that the buyer can lose and the maximum amount that the writer of the agreement can gain. Like a standard option, the only party that is required to perform in an interest-rate agreement is the writer. Once again like a standard call option contract, the buyer of an interest-rate cap benefits if the underlying interest rate rises above the strike rate because the seller (writer) must compensate the buyer. Like a standard put option contract, the buyer of an interest rate floor benefits if the interest rate falls below the strike rate, because the seller (writer) must compensate the buyer. To better understand interest-rate caps and floors, we can look at them as in essence equivalent to a package of interest-rate options. Because the buyer benefits if the interest rate rises above the strike rate, an interest-rate cap is similar to purchasing a package of call options on an interest rate or purchasing a package of put options on a bond. The seller of an interest-rate cap has effectively sold a package of call options on an interest rate or sold a package of put options on a bond. The buyer of an interest-rate floor benefits from a decline in the interest rate below the strike rate. Therefore, the buyer of an interest-rate floor has effectively bought a package of put options on an interest rate or a package of call options on a bond from the writer of the option. 17. How can an interest-rate collar be created? Interest-rate caps and floors can be combined to construct an interest-rate collar. This is achieved by buying an interest-rate cap and selling an interest-rate floor. By buying a cap and selling a floor, a financial institution can create a predetermined range for its cost of funds (i.e., a collar). 18. Value a three-year interest rate floor with a $10 million notional amount and a floor rate of 4.8% using the binomial interest-rate trees shown in Exhibit 28-11. The arbitrage-free binomial model described in Chapter 18 can also be used to value a floor and a cap. This is because a floor and a cap are nothing more than a package or strip of options. More specifically, they are a strip of European options on interest rates. To value a floor, the value of each period’s floor, called a floorlet, is found and all the floorlets are then summed. We refer to this approach to valuing a floor as the floorlet method. (The same approach can be used to value a cap.) For our floorlet problem, we use the binomial interest-rate tree (used in Chapter 18 to value an interest rate option) to value a 4.8%, three-year floor with a notional amount of $10 million. The reference rate is the one-year rates in the binomial tree in Exhibit 28-11. The payoff for the floor is annual. There is one wrinkle having to do with the timing of the payments for a cap and floor that requires a modification of the binomial approach presented to value an interest rate option. This is due to the fact that settlement for the typical cap and floor is paid in arrears. This means that the interest rate paid on a financial instrument with this feature is determined at the beginning of the period but paid at the end of the period (i.e., beginning of the next period). The text uses Arabic numbers for dates (i.e., Date 0, Date 1, Date 2, and Date 3) and words for years (i.e., Year One, Year Two, and Year Three) to avoid confusion regarding when a payment is determined and when it is made. Exhibit 28-11 shows the binomial interest rate tree with dates and years. Using Exhibit 28-11 we can understand what is meant by payment in arrears. Consider Date 1. There are two interest rates, 4.4448% and 5.4289%. Suppose today, Date 0, that an investor enters into an agreement whereby if a reference rate at Date 1 is below 4.8%, the investor will receive at Date 2 the difference between the reference rate and 4.8% multiplied by $10 million; if the reference rate is greater than 4.8% nothing is received. This agreement specifies payment in arrears. For example, consider at Date 1 the interest rate (reference rate) of 4.4448%. Here there would be a payoff of: (4.8% – 4.4448%) × $10,000,000 = $35,520.00. The payment of $35,520.00 would be made at Date 2. With this background, we will use the floorlet method to value the three-year floor. Consider first the value of the floorlet for Year One. At Date 0, the one-year rate is 3.5%. Since it is below the floor rate of 4.8%, there would be a payoff of: (4.8% – 3.5%) × $10,000,000 = $130,000. The present value is $130,000/1.035 = $125,603.86. This value of $125,603.86 occurs at Date 0 and is the Value of Year One floorlet. We now move on to the Year Two floorlet. There are two interest rates at Date 1: 4.4448% and 5.4289%. If the interest rate is 4.4448% on Date 1, there is a payoff as explained earlier. The payoff is $35,520.00 and will be made at Date 2. If the interest rate is 5.4289%, there is no payoff because the rate is greater than 4.8%. The payoff at Date 2 is either $35,520.00 or $0. These values have to be discounted back to Date 0. The discounting requires first discounting back to Date 1 and then discounting back to Date 0. At each date, the present values are averaged because of the assumption that both payoffs are equally likely. The discount rate used is the one at the node at the date where the interest rate is to be discounted back to. For example, the payoff of $35,520.00 at Date 2 should be discounted at 4.4448% to get the present value at Date 1. The present value is $35,520/1.044448 = $34,008.39 and occurs at Date 1. Obviously, the present value of the $0 payoff is $0. The average of these two present values at Date 1 is $17,004.20. The present value at Date 1 is then discounted back to Date 0 by using 3.5%. The present value is $17,004.20/1.035 = $16,429.18. This value of $16,429.18 occurs at Date 0 and is the Value of Year Two floorlet. For the valuation of the Year Three floorlet, there are three interest rates shown at Date 2. They are: 4.6958%, 5.7354%, and 7.0053%. There is no payoff if the interest rate at Date 2 is either 5.7354% or 7.0053% since these rates are greater than 4.80%. For the other interest rates, the payoff is: (4.8% – 4.6958%) × $10,000,000 = $10,420.00 This payoff is shown at Date 3 because it is paid in arrears. The present value of this payoff on Date 2 to be received at Date 3 is: $10,420.00/1.046958 = $9,952.64. This present value is shown at Date 2. Moving backwards to Date 1, we have to average the two values at Date 2 and discount back at the corresponding interest rate. Because the other value is zero, the average value at Date 2 is ($9,952.64 + $0)/2 = $4,976.32. Because the lower interest rate at Date 1 is 4.4448%, the present value at Date 1 is therefore $4,976.32/1.044448 = $4,764.55 and would be shown at the node at Date 1. The final step is to discount back to Date 0 the value of the payoff at Date 1. Because the other payoff is zero, the average of the two payoffs at Date 1 is $2,382.27. Discounting at 3.5% gives the value of the Year Three floorlet of $2,301.71. This value of $2,301.71 occurs at date 0 and is the Value of Year Three floorlet. The value of the three-year interest rate floor is the sum of the three floorlets given as: value of floor = value of Year One floorlet + value of Year Two floorlet + value of Year Three floorlet. For our floor floorlet problem, we get: Value of Year One floorlet: $125,603.86 Value of Year Two floorlet: $16,429.18 Value of Year Three floorlet: $2,301.71 Three-year interest rate floor: $144,334.75 Thus, to the nearest dollar, the value of the floor is $144,335. Solution Manual for Bond Markets, Analysis and Strategies Frank J. Fabozzi 9780132743549, 9780133796773