Chapter 18
Using Derivatives in Fixed Income Portfolio Management

In the previous chapter our focus was on the application of derivatives to the management of an equity portfolio. In this chapter, we look at how fixed income portfolio managers can use interest rate derivatives (Treasury futures, Treasury futures options, and interest rate swaps) to control interest rate risk and how credit derivatives (in particular credit default swaps) can be used to control credit risk.

18.1 Controlling Interest Rate Risk Using Treasury Futures

There are several interest rate futures contracts that are available to portfolio managers. The contracts commonly used to control the interest rate risk of bond portfolios are Treasury bond and note futures contracts. We describe these contracts here, their pricing in terms of deviations from the basic futures pricing model in Chapter 16, and how they are used.

The Treasury bond and note futures contracts are traded on the Chicago Mercantile Exchange (CME). The underlying instrument for the bond futures contract is $100,000 par value of a hypothetical 20-year, 6% coupon bond. This hypothetical bond's coupon rate is called the notional coupon. There is also an ultra-Treasury-bond futures contract. An acceptable deliverable for the ultra-Treasury-bond futures contract is a bond with a maturity of at least 25 years, allowing bond portfolio managers to better manage the longer end of the yield curve. The Treasury bond futures contract is effectively a short maturity bond futures contract and the ultra-Treasury-bond futures contract is effectively a long maturity bond futures contract with a duration that is better suited for managing duration for portfolios of long-term bonds. There are three Treasury note futures contracts: 10-year, 5-year, and 2-year. All three contracts are modeled after the Treasury bond futures contract and are traded on the CME. The underlying instrument for the 10-year Treasury note contract is $100,000 par value of a hypothetical 10-year, 6% Treasury note. Treasury futures contracts trade with March, June, September, and December settlement months. The futures price is quoted in terms of par being 100. Because the bond and notes futures contracts are similar, for the remainder of our discussion we focus on the bond futures contract.

We have been referring to the underlying instrument as a hypothetical Treasury bond. Whereas some interest rate futures contracts can only be settled in cash, the sellers of Treasury bond futures contracts must deliver some Treasury bond at settlement, unless they choose to liquidate their position prior to the settlement date by buying back the contract. This begs the question “which Treasury bond?” The CME allows the seller to deliver one of several Treasury bonds that the CME specifies are acceptable for delivery. Traders who are short a particular bond are always concerned about the risk of being unable to obtain sufficient securities to cover their position.

The bond issues that meet the delivery requirements for a particular contract are referred to as deliverable issues. The CME selects the Treasury issues that are acceptable for delivery from all outstanding Treasury issues that have at least 15 years to maturity from the first day of the delivery month. For settlement purposes, the CME specifies that a given issue's term to maturity is calculated in complete three-month increments (that is, complete quarters). For example, if the actual maturity of an issue is 15 years and 5 months, it would be rounded down to a maturity of 15 years and 1 quarter (three months). Moreover, all bonds delivered by the seller must be of the same issue.

Keep in mind that, while the underlying Treasury bond for this contract is a hypothetical issue and therefore cannot itself be delivered into the futures contract, the bond futures contract is not a cash settlement contract. The only way to close out a Treasury bond futures contract is either to initiate an offsetting futures position or to deliver one of the deliverable issues.

The delivery is as follows. On the settlement date, the seller of the futures contract (the short) is required to deliver to the buyer (the long) $100,000 par value of a 6%, 20-year Treasury bond. As noted, no such bond exists, so the short must choose one of the deliverable issues to deliver to the long. Suppose the seller selects a 5% coupon, 20-year Treasury bond to settle the futures contract. Because the coupon of this bond is less than the notional coupon of 6%, this would be unacceptable to the buyer who contracted to receive a 6% coupon, 20-year bond with a par value of $100,000. Alternatively, suppose the seller is compelled to deliver a 7% coupon, 20-year bond. Since the coupon of this bond is greater than the notional coupon of 6%, the seller would find this unacceptable. How does the exchange adjust for the fact that deliverable issues have coupons and maturities that differ from the notional coupon of 6%?

To make delivery equitable to both parties, the CME publishes conversion factors for adjusting the price of each deliverable issue for a given contract. Given the conversion factor for a deliverable issue and the futures price, the adjusted price is found by multiplying the conversion factor by the futures price. The adjusted price is called the converted price. That is,

equation

For example, suppose the settlement price of a Treasury bond futures contract is 110 and the deliverable issue selected by the short has a conversion factor of 1.25. Given the contract size is $100,000, the converted price for the deliverable issue is c18-math-0002.

The price that the buyer must pay the seller when the deliverable issue is delivered is called the invoice price. The invoice price is the converted price plus the deliverable issue's accrued interest. That is, the invoice price is:

equation

In selecting the issue to be delivered, the short will select from all the deliverable issues the one that will give the largest rate of return from a cash-and-carry strategy. We explained this strategy in Chapter 16 where we derived the theoretical futures price. In the case of Treasury bond futures, a cash-and-carry strategy is one in which a cash bond that is acceptable for delivery is purchased with borrowed funds and simultaneously the Treasury bond futures contract is sold. The bond purchased can be delivered to satisfy the short futures position. Thus, by buying the Treasury issue that is acceptable for delivery and selling the futures, an investor has effectively sold the bond at the delivery price (that is, the converted price).

A rate of return can be calculated for this strategy. This rate of return is referred to as the implied repo rate. Once the implied repo rate is calculated for each deliverable issue, the issue selected for delivery will be the one that has the highest implied repo rate (that is, the issue that gives the maximum return in a cash-and-carry strategy). The issue with the highest return is referred to as the cheapest-to-deliver issue (CTD issue). This issue plays a key role in the pricing of a Treasury futures contract.1

In addition to the choice of which acceptable Treasury issue to deliver—sometimes referred to as the quality option or the swap option—the short has at least two more options granted under CME delivery guidelines. The short is permitted to decide when in the delivery month, delivery actually will take place. This is called the timing option. The other option is the right of the short to give notice of intent to deliver up to 8:00 P.M. Chicago time after the closing of the exchange (3:15 P.M. Chicago time) on the date when the futures settlement price has been fixed. This option is referred to as the wildcard option. The quality option, the timing option, and the wildcard option—in sum referred to as the delivery options—mean that the long can never be sure which Treasury bond issue will be delivered or when it will be delivered.

18.1.1 Strategies for Controlling Interest Rate Risk with Treasury Futures

Portfolio managers can use interest rate futures to alter the interest rate sensitivity, or duration, of a portfolio. Those with strong expectations about the direction of the future course of interest rates will adjust the duration of their portfolios so as to capitalize on their expectations. Specifically, a portfolio manager who expects rates to increase will shorten duration; a portfolio manager who expects interest rates to decrease will lengthen duration. While portfolio managers can use cash market instruments to alter the duration of their portfolios, using futures contracts provides a quicker and less expensive means for doing so (on either a temporary or permanent basis).

A formula to approximate the number of futures contracts necessary to adjust the portfolio duration to some target duration is2

18.1 equation

The dollar duration of the futures contract is the dollar price sensitivity of the futures contract to a change in interest rates.

Notice that if the portfolio manager wishes to increase the portfolio's current duration, the numerator of the formula is positive. This means that futures contracts will be purchased. That is, buying futures increases the duration of the portfolio. The opposite is true if the objective is to shorten the portfolio's current duration: the numerator of the formula is negative and this means that futures must be sold. Hence, selling futures contracts reduces the portfolio's duration.

Hedging is a special case of risk control where the target duration sought is zero. If cash and futures prices move together, any loss realized by the hedger from one position (whether cash or futures) will be offset by a profit on the other position. As we explained in Chapter 17.1.3.2, when the net profit or loss from the positions is exactly as anticipated, the hedge is referred to as a perfect hedge. Hedging in bond portfolio management is more complicated than the examples with stock index futures we gave in Chapter 17.1.3.2. In bond portfolio management, typically the bond to be hedged is not identical to the bond underlying the futures contract and therefore there is cross-hedging. This may result in substantial basis risk.

Conceptually, cross-hedging is more complicated than hedging deliverable securities because it involves two relationships. In the case of Treasury bond futures contracts, the first relationship is between the CTD issue and the futures contract. The second is the relationship between the security to be hedged and the CTD issue.

The key to minimizing risk in a cross-hedge is to choose the right hedge ratio. The hedge ratio depends on volatility weighting, or weighting by relative changes in value. The purpose of a hedge is to use gains or losses from a futures position to offset any difference between the target sale price and the actual sale price of the security.

Accordingly, the hedge ratio is chosen with the intention of matching the volatility (that is, the dollar change) of the Treasury bond futures contract to the volatility of the bond to be hedged. Consequently, the hedge ratio for a bond is given by

18.2 equation

For hedging purposes, we are concerned with volatility in absolute dollar terms. To calculate the dollar volatility of a bond, one must know the precise time that volatility is to be calculated (because volatility generally declines as a bond moves toward its maturity date), as well as the price or yield at which to calculate volatility (because higher yields generally reduce dollar volatility for a given yield change). The relevant point in the life of the bond for calculating volatility is the point at which the hedge will be lifted. Volatility at any other point is essentially irrelevant because the goal is to lock in a price or rate only on that particular day. Similarly, the relevant yield at which to calculate volatility initially is the target yield. Consequently, the “volatility of bond to be hedged” referred to in equation (18.3) for the hedge ratio is the price value of a basis point for the bond on the date the hedge is expected to be delivered. The price value of a basis point (PVBP) for a bond is computed by changing the yield of a bond by one basis point and determining the change in the bond's price. It is a measure of bond price volatility to interest rate changes and is related to duration.

To calculate the hedge ratio given by equation (18.3), the portfolio manager needs to know the volatility of the Treasury futures contract. Fortunately, knowing the volatility of the bond to be hedged relative to the CTD issue and the volatility of the CTD bond relative to the futures contract, we can modify the hedge ratio given by equation (18.2) as follows:

18.3 equation

The second ratio above can be shown to equal the conversion factor for the CTD issue. Assuming a fixed yield spread between the bond to be hedged and the CTD issue, equation (18.3) can be rewritten as

18.4 equation

where PVBP is equal to the price value of a basis point.

Given the hedge ratio, the number of contracts that must be short is determined as follows:

18.5 equation

For example, suppose that the amount to be hedged is $20 million and a Treasury bond futures contract is used for hedging. The par value for a Treasury bond futures contract is $100,000. The ratio in the above equation is then 200 (= $20,000,000/$100,000), which means that the number of futures contracts that must be sold for a bond to be hedged is 200 contracts.

18.1.2 Pricing of Treasury Futures

In Chapter 16.2.3, we explained the basic pricing model for a generic futures contract. We also explained how the model must be modified for stock index futures in Chapter 17.1.2. Let's look at how the specifics of the Treasury futures contract necessitate the refinement of the theoretical futures pricing model. The assumptions that require a refinement of the model are the assumptions that (1) there are no interim cash flows and (2) the deliverable asset and the settlement date are known.

With respect to interim cash flows, for a Treasury futures contract the underlying is a Treasury note or a Treasury bond. Unlike a stock index futures contract, the timing of the interest payments that will be made by the U.S. Department of the Treasury for every issue that is acceptable as deliverable for a contract is known with certainty and can be incorporated into the pricing model. However, the reinvestment interest that can be earned from the coupon payment from the payment dates to the settlement date of the contract is unknown and depends on prevailing interest rates at each payment date.

Now let's look at the implications of the assumption that there are a known deliverable and a known settlement date. Neither assumption is consistent with the delivery rules for some futures contracts. For U.S. Treasury note and bond futures contracts, for example, the contract specifies deliverable issues that can be delivered to satisfy the contract. Although the party that is long the contract (that is, the buyer of the contract) does not know the specific Treasury issue that will be delivered, the long can determine the CTD issue from among the deliverable issues. It is this issue that is used in obtaining the theoretical futures price. The net effect of the short's option to select the issue to deliver to satisfy the contract is that it reduces the theoretical future price by an amount equal to the value of the delivery option granted to the short.

Moreover, unlike other futures contracts, the Treasury bond and note contracts do not have a delivery date. Instead, there is a delivery month. The short has the right to select when in the delivery month to make delivery. The effect of this option granted to the short is once again to reduce the theoretical futures price. More specifically,

18.6 equation

18.2 Controlling Interest Rate Risk Using Treasury Futures Options

Interest rate options can be written on a fixed income security or an interest rate futures contract. The former options are called options on physicals and the latter are called futures options. The most liquid exchange-traded option on a fixed income security is an option on Treasury futures traded on the CME. Goodman (1985) gives three reasons why futures options on Treasuries are preferred to options on physicals as the options vehicle of choice for institutional investors:

  1. Unlike options on Treasury securities, options on Treasury futures do not require payments for accrued interest to be made. Consequently, when a futures option is exercised, the call buyer and the put writer need not compensate the other party for accrued interest.
  2. Futures options are believed to be “cleaner” instruments because of the reduced likelihood of delivery squeezes. Market participants who must deliver a Treasury security are concerned that at the time of delivery, the Treasury to be delivered will be in short supply, resulting in a higher price to acquire the security. Because the deliverable supply of futures contracts is more than adequate for futures options currently traded, there is no concern about a delivery squeeze.
  3. In order to price any option, it is imperative to know at all times the price of the underlying instrument. In the bond market, current prices are not as easily available as price information on the futures contract. The reason is that Treasury securities trade in the OTC market and, consequently, there is less price information compared to Treasury futures which are traded on an exchange.

However, portfolio managers do make use of over-the-counter (OTC) options. Typically they are purchased by institutional investors who want to hedge the risk associated with a specific security or index. Besides options on fixed income securities, there are OTC options on the shape of the yield curve or the yield spread between two securities. A discussion of these OTC options is beyond the scope of this chapter. Because the most common option used in bond portfolio management is the exchange-traded futures option on Treasury securities, we limit our discussion to this derivative.

A futures option gives the buyer the right to buy from or sell to the writer a designated futures contract at the strike price. If the futures option is a call option, the buyer has the right to purchase one designated futures contract at the strike price. That is, the buyer has the right to acquire a long futures position in the designated futures contract. If the buyer exercises the call option, the writer acquires a corresponding short position in the futures contract.

A put option on a futures contract grants the buyer the right to sell one designated futures contract to the writer at the strike price. That is, the option buyer has the right to acquire a short position in the designated futures contract. If the put option is exercised, the writer acquires a corresponding long position in the designated futures contract. There are futures options on all the Treasury bond and note futures contracts. A summary of the rights of the buyer and the obligation of the seller and what the payoff is if the buyer exercises follows:

  • Call buyer has the right to purchase one futures contract at the strike price.
    • If exercised, the seller has a short futures position.
    • If exercised, the seller pays the buyer (current futures price – strike price).
  • Put buyer has the right to sell one futures contract at the strike price.
    • If exercised, the seller has a long futures position.
    • If exercised, the seller pays the buyer (strike price – current futures price).

The CME's Treasury bond futures contracts have delivery months of March, June, September, and December. As with stock index futures contracts, there are flexible Treasury futures options. 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 CME constraints.

Because 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 strike 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 call futures option, the option writer must pay to the buyer of the option the difference between the current futures price and the strike price. In the case of a put futures option, the option writer must pay the option buyer the difference between the strike price and the current futures price.

For example, suppose an investor buys a call option on some futures contract in which the strike price is $140. Assume also that the futures price is $150 and that the buyer exercises the call option. Upon exercise, the call buyer is given a long position in the futures contract at $140, and the call writer is assigned the corresponding short position in the futures contract at $140. The futures positions of the buyer and the writer are immediately marked to market by the exchange. Because the prevailing futures price is $150 and the strike price is $140, the long futures position (the position of the call buyer) realizes a gain of $10, while the short futures position (the position of the call writer) realizes a loss of $10. The call writer pays the exchange $10, and the call buyer receives $10 from the exchange. The call buyer, who now has a long futures position at $150, can either liquidate the futures position at $150 or maintain a long futures position. If the former course of action is taken, the call buyer sells a futures contract at the prevailing futures price of $150. There is no gain or loss from liquidating the position. Overall, the call buyer realizes a gain of $10. The call buyer who elects to hold the long futures position will face the same risk and reward of holding such a position, but still realizes a gain of $10 from the exercise of the call option.

Suppose, instead, that the futures option is a put rather than a call, and the current futures price is $125 rather than $150. If the buyer of this put option exercises it, the buyer would have a short position in the futures contract at $140; the option writer would have a long position in the futures contract at $140. The exchange then marks the position to market at the then-current futures price of $125, resulting in a gain to the put buyer of $15, and a loss to the put writer of the same amount. The put buyer, who now has a short futures position at $125, can either liquidate the short futures position by buying a futures contract at the prevailing futures price of $15 or maintain the short futures position. In either case, the put buyer realizes a gain of $15 from exercising the put option.

18.2.1 Strategies for Controlling Interest Rate Risk Using Treasury Futures Options

In our review of the use of options in equity portfolio management in Chapter 17.2.2, we explained how they can be used for risk management, return enhancement, and cost management. We do not repeat the explanation of the applications here. Instead, we explain how futures options can be used for hedging and return enhancement. More specifically, we illustrate a protective put strategy (a risk management application) and a covered call writing strategy (a return enhancement application). As will be seen, the applications are complicated by the fact that the option is not an option on a physical but a futures option.

Buying puts on Treasury futures is one of the easiest ways to purchase protection against rising rates. As explained in Chapter 17, this strategy is called a protective put strategy. We also explained the technical aspects of implementing this strategy for stock index futures. Here we explain the complexities associated with implementing this strategy where what is to be protected is an individual bond issue.

In our explanation of the process, we assume that the bond issue to be protected is an investment grade non-Treasury security. The reason for the credit quality being investment grade is that we are protecting against an adverse movement in interest rates. The lower the investment-grade rating of a bond, the more of its price depends on its equity component.

This protective put strategy is a cross-hedge because the bond to be hedged and the underlying for the Treasury options futures contract are not the same. There are many candidate Treasury options futures contracts that can be employed as the hedging instrument. In the explanation of the process, we see how we need to use the CTD issue to implement the strategy.

Given the bond issue to be hedged and the particular Treasury options futures contracts that are used, the process involves the following steps:

  1. Step 1: Determine the minimum price for the bond to be hedged. If there was a put option on the bond to be hedged, then this would be its strike price.
  2. Step 2: Given the minimum price for the bond to be hedged, the yield for that bond can be computed. This is simple. It involves computing the yield given the minimum price determined in Step 1. This gives us the minimum yield for the bond to be hedged. Now we need to link this minimum yield to get the strike price for the Treasury options futures contract.
  3. Step 3: Given the minimum yield for the bond to be hedged, the minimum yield on the CTD issue can be determined. Remember that the bond to be hedged is a non-Treasury security and it will trade in the market at a higher yield than the CTD issue, which is a Treasury security. To determine the minimum yield on the CTD issue, a credit spread must be assumed. This credit spread is typically found by using a simple linear regression.3 What is important to bear in mind is that the strategy is dependent on this assumed relationship (i.e., the assumed credit spread). Once this step is completed, the resulting credit spread added to the minimum yield for the bond to be hedged gives the minimum yield for the CTD issue.
  4. Step 4: Given the minimum yield for the CTD issue found in Step 3 and the coupon rate and maturity of the CTD issue, the target price for the futures option can be determined using the basic yield-price calculation; that is, given a bond's coupon, principal (which is assumed to be 100), and maturity, the price can be determined.4 This price for the CTD issue corresponds to the minimum price for the bond to be hedged found in Step 1.
  5. Step 5: Given the minimum price for the CTD issue, the strike price for the futures option is calculated. Remember there are several candidate futures contracts for a given expiration date over which the hedge is sought. For each one, the corresponding strike price is found by multiplying the minimum price for the CTD issue by the conversion factor for the associated futures option.

The five steps described above are always necessary to identify the appropriate strike price on a put futures option. The process involves estimating

  • The relationship between price and yield.
  • The assumed relationship between the yield spread between the bonds to be hedged and the CTD issue.
  • The conversion factor for the CTD issue.

As with hedging with futures, explained earlier in this chapter, the success of the hedging strategy will depend on (1) whether the CTD issue changes and (2) the yield spread between the bonds to be hedged and the CTD issue.

The hedge ratio is determined using equation (18.5) because we will assume a constant yield spread between the bond to be hedged and CTD issue. To compute the hedge ratio, the portfolio manager must calculate the price value of a basis point for the CTD issue and the bond to be hedged at the option expiration date and at the yields corresponding to the futures' strike price of the yield. To obtain the number of put futures options that should be purchased for the put protective strategy, equation (18.6) uses “par value of the futures options contract” instead of “par value of the futures contract.”

18.2.2 Pricing Models for Treasury Futures Options

In Chapter 16.3.2.2, we discussed the Black-Scholes option pricing model. In this section, we provide an overview of pricing models for options on Treasury futures options. In general, these options are much more complex than options on stocks or stock indexes because of the need to take into consideration the term structure of interest rates.

The most commonly used model for pricing futures options is the one developed by Black (1976). The Black model was initially developed for valuing European options on forward contracts. There are two problems with this model. The first is the assumptions about the underlying asset for the futures contract. Specifically, there are three unrealistic assumptions:

  1. The probability distribution for the prices assumed permits some probability—no matter how small—that the price can take on any positive value. But, unlike stock prices, Treasury futures prices (the underlying in the case of a Treasury futures contract) have a maximum value. So any probability distribution for the Treasury futures prices assumed by an option pricing model that permits prices to be higher than the maximum value could generate nonsensical option prices.
  2. In the pricing model for futures options it is assumed that the short-term interest rate is constant over the life of the option. Yet the price of a Treasury futures contract 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 pricing Treasury futures options.
  3. The volatility (standard deviation) of futures 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 volatility is constant over the life of a Treasury futures option is inappropriate.

The second problem is that the Black model was developed for pricing European options on forward contracts. Treasury futures options, however, are American options. This problem can be overcome. The Black model was extended by Barone-Adesi and Whaley (1987) to American options on futures contracts. This is the model used by the CME to settle the flexible Treasury futures options. However, this model was also developed for equities and is subject to the first problem noted above. Despite its limitations, the Black model is the most popular option pricing model for options on Treasury futures.

18.3 Controlling Interest Rate Risk Using Interest Rate Swaps

In its most basic form, an interest rate swap is an agreement between two parties to exchange cash flows periodically. In a plain-vanilla swap, over the life of the contract one party pays a fixed rate of interest based on a notional amount in exchange for a floating rate of interest based on the same notional amount from the counterparty. Typically, no principal is exchanged at the beginning or end of a swap.

The fixed rate on a swap is ordinarily set at a rate such that the net present value of the swap's cash flows is zero at the start of the swap contract. This fixed rate is called the swap rate. The difference between the swap rate and the yield on an equivalent-maturity Treasury is called the swap spread.5 The floating rate on a swap is typically benchmarked off the London Interbank Offered Rate (LIBOR) or constant maturity Treasury (CMT) rate. In a plain-vanilla swap, the floating rate is three-month LIBOR, which resets and pays quarterly in arrears. Although different types of interest rate swaps exist, including vanilla swaps, basis swaps, indexed-amortizing swaps, and callable swaps, to name a few, here our focus is on generic interest rate swaps as hedge instruments.

The party to a swap that pays the swap spread is referred to as the fixed-rate payer or, equivalently, the floating-rate receiver. The party that receives the fixed-rate is referred to as the fixed-rate receiver or, equivalently, the floating-rate payer. Here we simply refer to the two parties as the fixed-rate payer and fixed-rate receiver.

In Chapter 16, we explained why an interest rate swap can be interpreted as a package of forwards or futures contracts. Another way of thinking about an interest rate swap from the prospective of the fixed-rate receiver is that it is equivalent to a position that is long a fixed-rate bond position (i.e., owning a fixed-rate bond), which is completely financed at the short-term interest rate that is the reference rate for the swap. In a completely financed long bond position, fixed-rate interest payments are received and floating rate financing costs are paid periodically, with the final principal payment from the bond used to repay the initial financing of the bond purchase. On a net basis, a completely financed bond position has zero cost, like a swap, and the periodic cash flows replicate the cash flows on a swap. In fact, a swap from the perspective of the fixed-rate receiver is a fully leveraged bond position where the financing rate is equivalent to the swap's reference rate. Hence, this alternative view of an interest rate swap from the perspective of the fixed-rate receiver is appealing because it implies that a swap can be used as an alternative hedging vehicle to Treasury futures contracts to manage interest rate risk. The position of a fixed-rate receiver is equivalent to shorting a fixed-rate bond and investing the proceeds in a floating-rate bond. Once again, this is appealing because it suggests that swaps can be used as an alternative instrument to manage interest rate risk.

18.3.1 Strategies for Controlling Interest Rate Risk Using Interest Rate Swaps

To control a portfolio's interest rate risk using an interest rate swap, it is necessary to understand how the value of a swap changes as market interest rates change. As explained in Chapter 13.4.1.1, the measure used to quantify the change in the dollar value of a fixed-income instrument to changes in interest rate is dollar duration. Therefore, it is necessary to determine the dollar duration of an interest rate swap in order to implement effectively a risk control strategy with this derivative.

As explained above, from the perspective of a fixed-rate receiver, the interest-rate swap position can be viewed as long a fixed-rate bond plus short a floating-rate bond. 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,

18.7 equation

Let's look at the relative magnitude of the two components in equation (18.7). Consider first the dollar duration of a floating-rate bond. The dollar duration will depend on what the length of time to the reset date is. The shorter this length of time, the smaller the dollar duration of a floating-rate bond is. Since in a typical interest-rate swap the time to the next reset date is very short, the dollar duration of a floating-rate bond will be small. In contrast, the dollar duration of a fixed-rate bond will be considerably greater. Thus, a good approximation of the dollar duration of an interest-rate swap is the dollar duration of a fixed-rate bond.

The implication here is that if a portfolio manager wants to increase the dollar duration of a portfolio using an interest-rate swap, the portfolio manager should take a position as a fixed-rate receiver. This is economically equivalent to leveraging a portfolio's interest-rate exposure by adding dollar duration. By entering into an interest-rate swap as the fixed-rate payer, instead, the portfolio manager reduces the portfolio's dollar duration.

Suppose that a portfolio manager has a target duration for the portfolio. That target duration can be used to obtain the target dollar duration for the portfolio. The target portfolio dollar duration is the sum of the current dollar duration of the bond portfolio and the dollar duration of the interest-rate swap. That is,

18.8 equation

Solving for the dollar duration of the swap,

18.9 equation

For example, consider a $100 million bond portfolio that has a current duration of 5 and the portfolio manager wants to increase the portfolio duration to 6 (i.e., 6 is the target duration). We know that for a 100 basis point change in interest rates, the current portfolio value will change by 5%. Therefore, the dollar duration is $5 million (=5% × $100,000,000). Similarly, the change in the portfolio value that the portfolio manager seeks for a target duration of 6 is 6% and the dollar duration is $6 million. Therefore, from equation (18.9), using interest-rate swaps the portfolio manager must add to the portfolio $1 million in dollar duration. The dollar duration of a swap contract can be determined by changing interest rates by 100 basis points and computing the average change in the value of the swap.

In our example, to increase the portfolio dollar duration using an interest-rate swap, the portfolio must be the counterparty that is a fixed-rate receiver. Suppose instead that the portfolio manager seeks to reduce the target duration to 4.5. In this case, the target dollar duration is $4.5 million and from equation (18.9), the dollar duration of a swap to achieve the target duration is to reduce the dollar duration by $500,000. The portfolio manager should then enter into a swap as the fixed-rate payer.

18.3.2 Pricing of Interest Rate Swaps

Although there is a wide variety of interest rate swaps available to a portfolio manager to control interest rate risk, the main idea when pricing all of them is that the fair value of a swap should be the difference between the present values of the expected cash flows exchanged between the two parties in the swap. Again, we focus on the generic interest rate swap.

In a generic interest rate swap, the cash flows on the fixed component (i.e., the fixed-rate payments) are known at the inception of the swap. However, the future cash flows on the floating component are unknown because they depend on the future value of the reference rate. The future floating rates for purposes of valuing a swap are derived from forward rates that are embedded in the current yield curve.6 By utilizing forward rates, a swap net cash flow can be derived throughout the life of a swap. The sum of these cash flows discounted at the corresponding forward rate for each time period is the current value of the swap. Mathematically, the value of a swap position is:

18.10 equation

where c18-math-0014 denotes the present value of x and c18-math-0015 are the dates at which payments are made. The fair swap rate is the fixed rate that makes the swap value zero.

An alternative approach to pricing a generic interest rate swap is to view it as two simultaneous bond payments made by the two parties. Namely, think of the fixed-rate payer as paying the notional amount to the fixed-rate receiver at the termination date, and of the fixed-rate receiver as paying the notional amount to the fixed-rate payer at the termination date. This slight modification does not change the actual cash flows and value of the swap, because the payments of the notional amounts cancel out at the termination date. However, it does help us imagine the stream of payments from the fixed-rate payer as the value of a fixed-coupon bond, and the stream of payments from the fixed-rate receiver as the value of a floating-rate bond.

Let the notional amounts be 100, and let ν denote the premium (per annum) paid by the fixed-rate payer. Assume that the payments happen at dates 1, 2, …, T, and that the time interval between payments is Δt. (This time interval is typically a quarter.)

At time 0, the value of the fixed-rate bond is

18.11 equation

where B(0,t) denotes the value (at time 0) of a zero-coupon bond with a face value of 1 and maturity t. (This is because the collection of payments during the life of the swap can be thought of as a portfolio of zero-coupon bonds of face value 1 with maturities equal to the times of the swap payments.)

The value of the floating-rate bond at time 0 is 100. To see this, note that the fixed-rate receiver can replicate the value of the bond by investing 100 today at the current interest rate, and earning just enough interest to pay the first coupon on the floating-rate bond to the fixed-rate payer. Then, the fixed-rate receiver can invest 100 again at the prevailing interest rates after the first swap payment, and earn enough interest to pay the second floating-rate coupon, with 100 left over. Continuing in the same way, the fixed-rate receiver can reinvest the 100 until the last time period, when he or she pays the 100 to the fixed-rate payer. Therefore, the present value of the investment from the perspective of the fixed-rate receiver is 100. From the perspective of the fixed-rate receiver, the value of the swap today is the difference between the fixed-rate payer's payments and the payments that must be made. That is,

18.12 equation

The fixed rate ν that makes the value of the swap equal to zero at time 0 is the fair price of the swap at time 0 and is the swap rate. It is easy to see that the value of ν (the swap rate) should be

18.13 equation

The values of c18-math-0019 can be determined from today's yield curve.7 They are in fact the discount factors that apply to different maturities.

18.4 Controlling Credit Risk with Credit Default Swaps

Thus far, we have discussed derivatives that can be used to control interest rate risk. Now we look at a derivative that can be used to control credit risk. The general category of derivatives to control credit risk is referred to as credit derivatives. The most commonly used type of credit derivative is the credit default swap (CDS) and for that reason we focus only on credit default swaps in this section.

There are two parties to a CDS: a credit protection buyer and a credit protection seller. The credit protection seller provides protection against some “credit event” for a periodic fee paid by the credit protection buyer. The CDS swap premium payment is the periodic payment made by the credit protection buyer to the credit protection seller.

The documentation of a CDS sets forth (1) the underlying for which the protection is being provided and (2) the specific credit event(s) for which protection is being provided. The underlying for which credit protection is being provided is a reference entity or a reference obligation. The reference entity is the issuer of the debt instrument for which credit protection is being sought. The reference obligation is the particular debt issue for which the credit protection is being sought. For example, a reference entity could be Exxon Mobile. The reference obligation would be a specific Exxon Mobile bond issue.

There are two types of CDS: single-name CDS and index CDS. As the name suggests, in a single-name CDS, there is only one reference entity or one reference obligation. In an index CDS, denoted by CDX, there is a standardized basket of reference entities. The two most actively traded CDXs on corporate bonds for reference entities in North America are the North America Investment Grade Index (CDX.NA.IG), which has 125 corporate reference entities that have an investment-grade rating and the North America High Yield Index (CDX.NA.HY) with 100 corporate entities that have a noninvestment-grade rating. For European corporate bonds, the most active CDX is the iTraxx Europe, which has 125 corporate reference entities. All three CDX use equal weighting of the reference entities. So for the CDX.NA.IG and the iTraxx Europe each reference entity is 0.8% of the index while for the CDX.NA.HY each reference entity is 1% of the index. The three index CDXs above are available in maturities from 1 to 10 years, with the greatest liquidity at 5-, 10-, and 7-year maturities.

The periodic swap premium payment made by the credit protection buyer to the credit protection seller is based on the swap rate and the notional amount. Consider first the swap premium payment for a single-name CDS (i.e., where the underlying is only one reference entity or reference obligation). Because typically the swap premium payments are quarterly, the payment is8

equation

For example, assume that the swap rate is 2.8% and the notional amount is $10 million. Then the quarterly swap premium payment is

equation

The quarterly swap premium payment over the life of a single-name CDS is the same each quarter.9 For an index CDS, however, the quarterly swap premium may decline because the notional amount each quarter may change. This is because an index CDS is written on a standardized basket of reference entities. If a credit event occurs for any of the reference entities in the index, then the notional amount is reduced accordingly as a result of the removal of those reference entities from the index.

18.4.1 Strategies for Controlling Credit Risk with Credit Default Swaps

As with other derivatives we have already discussed, to appreciate the potential application of a single-named CDS and an index CDS to control a portfolio's credit risk, it is helpful to look at the economic interpretation of these derivative products from the perspective of the counterparties. In our explanation we assume a single-name CDS where the reference obligation is bond XYZ issued by some corporation.

Consider an investor who purchases bond XYZ. The investor would make a cash outlay equal to bond XYZ's price, which we denote by c18-math-0022. Assuming that the issuer of bond XYZ does not default, the investor will receive semiannual cash inflows equal to one half of bond XYZ's annual coupon rate. The semiannual coupon payments will be received by the investor as long as the issuer of bond XYZ does not default. If the investor sells bond XYZ at time T, then there will be a cash inflow equal to bond XYZ's sale price. We denote this price by c18-math-0023. Suppose that at time T a credit event occurs that causes bond XYZ's price to fall below the purchase price paid by the investor (i.e., c18-math-0024). The investor then realizes a loss equal to the c18-math-0025.

Let's look at the cash flow for the credit protection seller in a single-name CDS where the reference obligation is bond XYZ. This party to the CDS receives a quarterly payment based on the CDS swap spread. That is, there is a cash inflow equal to the quarterly swap premium payment. However, the swap premium payments are made only if the issuer of bond XYZ does not trigger a credit event. Thus, as with an investor who buys bond XYX, there are periodic cash inflows as long as there is no adverse credit event that stops the payments (default in the case of owning the cash bond and credit event in the case of a CDS). It seems like this cash flow characteristic of the credit protection seller's position is similar to that of an investor who buys a cash bond.

Let's now suppose that a credit event occurs. The credit protection seller must make a payment to the credit protection buyer. This payment represents a cash outlay or loss for the credit protection seller. Yet, an investor in bond XYZ would also realize a loss if an adverse event occurs. Once again, this cash flow attribute is similar for both the credit protection seller and an investor in a bond.

Consequently, the credit protection seller has an economic position that is analogous to an investor in a cash bond. This is reasonable because both the credit protection seller and the investor who is long a cash bond are buyers of the bond issuer's credit risk. This interpretation is the same for an index CDS. Using the same cash flow analysis, it can be shown that the credit protection buyer is analogous to an investor who is short a bond in the cash market.

Now that we have the economic interpretation of a single-name CDS and an index CDS, we can appreciate how portfolio managers can use this derivative. CDS like many of the other derivatives described in this and the previous chapter is a more transactionally efficient vehicle for executing portfolios strategies and portfolio rebalancing. CDSs on corporate entities or specific obligations are generally more liquid than the underlying bonds. This makes it more efficient for portfolio managers to alter the exposure to one or more corporate bonds or a corporate bond index using CDS than transacting in the cash market.

Suppose that a bond portfolio manager wants to change credit exposure to either an individual corporate bond or a corporate bond index. To do so, the portfolio manager would sell protection (i.e., be the credit protection seller in a CDS) because as explained earlier this is analogous to a long position. Creating a long position in individual corporate names or a corporate bond index is often easier and less costly in the CDS market given its liquidity. A portfolio manager can reduce the exposure to a particular corporate issuer held in a portfolio by buying protection using a single-name CDS or reduce the exposure to a corporate bond index by using an index CDS. This is done by being the credit protection buyer.

18.4.2 General Principles for Valuing a Single-Name Credit Default Swap

In this section, we describe the general principles for valuing single-name CDSs on a corporate bond issuer.10 By valuing we mean determining the fair value of the CDS swap spread. As in valuing other derivatives, the general principle is that there is a relationship between the cash and the derivatives market.

As a reference obligation of the CDS, we consider a floating-rate debt obligation outstanding that has a maturity of T, is trading at par value, and offers a coupon rate of LIBOR plus a spread denoted by F. The coupon reset formula for this floating-rate debt obligation is equal to c18-math-0026.11 We assume that the CDS written on this floating-rate debt obligation requires physical delivery. To simplify further, we assume that the coupon payment dates for the floating-rate debt obligation are the same as the dates on which payments must be made to the credit protection seller of the single-name CDS.

Suppose that an investor has purchased the floating-rate debt obligation of the reference entity by using borrowed funds. The investor can do this by using a repurchase agreement (repo) and paying the repo rate.12 The repo rate is available for a time period equal to the maturity of the floating-rate debt obligation, which is T years. The borrowing rate for the repo (i.e., the repo rate) is LIBOR + B, where B is the spread over LIBOR, which is assumed to be constant over the repo's life.

In addition to the above, we make the following assumptions:

  • There is no counterparty risk with respect to the counterparty in the CDS and the counterparty in the repurchase agreement.
  • There are no transaction costs.
  • There is no difficulty in shorting bonds in the market.
  • Should a credit event occur, it does so one day following a coupon payment date.

The objective is to analyze how the premium for a single-name CDS with a maturity of T for some reference entity is determined. We denote this CDS premium by s. To do so, consider the following strategy:

  • The investor purchases the floating-rate debt obligation with maturity T issued by the reference entity.
  • The investor obtains the funds needed to purchase the floating-rate debt obligation by borrowing for the life of that debt obligation (which is also the term of the CDS), T, in the repo market.
  • The investor becomes a credit protection buyer, and hedges the credit risk associated with the floating-rate debt obligation by entering into a CDS with a maturity of T where the reference entity is the issuer of the floating-rate debt.

Let's look at the payoff for the two possible scenarios: no credit event occurs and a credit event occurs.

  • No-credit-event scenario: If no credit event occurs, then the floating-rate debt obligation matures. Over the life of the debt obligation, the interest earned is equal to LIBOR + F each period. The cost of borrowing (i.e., the repo rate) for each period is LIBOR + B. Hence, LIBOR + F is received from ownership of the floating-rate debt obligation and LIBOR + B is paid out to borrow funds. The net cash flow is therefore what is earned: c18-math-0027. Consequently, given the assumptions made above, the strategy will have a payoff of c18-math-0028 in the no-credit-event scenario.
  • Credit-event scenario: If a credit event occurs, there is physical delivery of the floating-rate debt obligation by the credit protection buyer to the credit protection seller. The credit protection seller then pays the full value of the floating-rate obligation to the credit protection buyer. By assumption, the credit event is assumed to occur right after the floating-rate debt obligation's coupon payment is made. Because the credit event's occurrence means that the CDS agreement is terminated, there are no further coupon payments and no accrued CDS payments. The proceeds obtained from the credit protection seller are used to repay the amount borrowed to purchase that security. As a result, the repo loan is repaid and the same payoff for the strategy as in the scenario where no credit event is realized (i.e., c18-math-0029).

Like the pricing arguments we presented in Chapter 16, a no-arbitrage requirement in this context means that the CDS spread, s, must be equal to the payoff under both scenarios, c18-math-0030. Thus, as a first approximation (because of the simplifying assumptions), the CDS spread is the difference between the spread over LIBOR at which the reference entity could issue a par floating-rate debt obligation c18-math-0031 and the spread over LIBOR to borrow funds in order to purchase that floating-rate debt obligation c18-math-0032.

As we said, this calculation is only an approximation. For example, one of the assumptions that should be noted is that the repo rate is constant over the repo's life, which is typically not the case. That is, one cannot borrow in the repo market at a fixed rate for several years. However, by entering into a CDS, one is effectively locking in a borrowing rate for the term of the CDS. This is the appeal of using a CDS rather than creating the same financing position with a repo. Another questionable assumption is that for corporate issuers that are reference entities for a single-name CDS, there is not likely to be a floating-rate debt obligation trading at par. More sophisticated CDS pricing models are available but they are beyond the scope of this book.13

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