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Trading and Capital-Markets Activities Manual

Instrument Profiles: Interest-Rate Swaps (Continued)
Source: Federal Reserve System 
(The complete Activities Manual (pdf format) can be downloaded from the Federal Reserve's web site)


Primary Market 

The primary market for interest-rate swaps consists of swap dealers, swap brokers, and end-users. 

Brokers and Dealers 

Financial institutions, such as commercial banks, investment banks, and insurance companies, act as dealers in interest-rate swaps. Banks are a natural intermediary in the swaps market because of their exposure to interest-rate movements and their expertise in analyzing customer credit risk. 

Swap brokers are paid a fee for arranging a swap transaction between two counterparties. Swap brokers do not take positions and do not act as a counterparty to a swap transaction. 


End-users of interest-rate swaps include financial institutions, corporations, sovereigns, government-sponsored enterprises (GSEs), and money managers. Banks who are dealers often also use swaps in an end-user capacity for asset/liability management, funding, and investment purposes. End-users use interest-rate swaps for hedging, investment, and speculative purposes. They also often use interest-rate swaps to reduce funding costs. 

The nature of an end-user's business often determines whether he or she will wish to be a fixed-rate receiver or a fixed-rate payer. Fixed-rate payers are often firms whose minimum cash flows are reasonably predictable regardless of the level of interest rates. This class includes manufacEagle Tradersg and distribution firms in the developed countries, financial institutions with large portfolios of fixed-rate assets, and national agencies of certain developed countries that have difficulty accessing fixed-rate funds. 

Fixed-rate receivers are often highly sensitive to changes in short-term market rates of interest. This class includes large money-center or regional banks that have large portfolios of floating-rate assets. The interest rates on the assets held in their loan portfolios may be indexed to U.S. prime rates, LIBOR, or other short-term market rates. The class also includes borrowers who have fixed-rate debt outstanding and prefer to convert it to floating-rate debt. Institutions such as life insurance companies, pension funds, wealthy investors, and managed trust accounts are notable examples of natural fixed-rate receivers. 

Secondary Market 

If a counterparty wishes to terminate, or unwind, an existing swap position in the secondary market, it must do so by one of three methods: swap reversal, swap assignment, or swap buyback (also called close-out or cancellation). 

In a swap reversal, a counterparty of a swap enters into an offsetting swap with the same terms as the original swap. For example, if Firm A is in a fixed-for-floating swap, paying 10 percent on $10 million notional for U.S. dollar three-month LIBOR, with one year to maturity, the offsetting swap would be a one-year floating for-fixed swap, paying U.S. dollar three-month LIBOR for 10 percent on $10 million notional. If market rates have changed since the position was initiated, which is likely, a mirror offsetting position cannot be established unless a fee is paid to establish the off-market mirror transaction. For instance, in the example above, if one-year rates at the time that the mirror swap is traded are 8 percent, the counterparty will have to pay a fee of approximately $185,000 to enter into the mirror trade ((10 percent - 8 percent) $10 million discounted at 8 percent). The counterparty does not cancel the first swap; it adds a second swap to its books at the cost of increasing default risk. 

In a swap assignment, a counterparty finds a new counterparty who is willing to assume its position in the swap. Swap assignments require the acquiescence of the other counterparty to the swap. At the time of the assignment, a payment representing the net present value of the swap is made either to or from the new assigned counterparty. For example, using the example above in which Firm A is in a 10 percent one-year fixed-for-floating swap, Firm A can assign its position in the swap to a new counterparty- Counterparty B (usually a dealer). In this case, as the swap has a negative mark-to-market value for Firm A, Firm A will be required to make a payment of $185,000 to Counterparty B. Counterparty B then assumes Firm A's position in the swap with the original counterparty. A key issue in swap sales is the creditworthiness of the firm or dealer who will assume the swap. If the creditworthiness is poor, the other counterparty may not agree to the sale. 

In a buy-back, one of the counterparties to a swap sells the swap to the other counterparty. Unlike the swap assignment example above, buy-backs are between the original counterparties and do not involve a third party. Buy-backs usually involve a payment which is based on the mark-to-market value of the swap at the time of the buy-back. In the example above, Firm A would be required to make a payment of $185,000 to the other original counterparty to terminate the swap. 

Market Transparency 

Market transparency in the swaps market is generally high. Market quotes are readily available on sources such as Telerate and Bloomberg. Increased competition has, in part, led to the narrowing of bid/offer spreads on plain vanilla deals. For instance, in the early 1980s, bid/offer spreads were in the 40 to 50 basis point range for deals under five years, and liquidity was almost nonexistent for deals beyond 10 years. Today, spreads have narrowed to 1 to 3 basis points for swaps under 10 years, and liquidity has increased significantly on swaps beyond 10 years. 

Liquidity in the secondary market is high but is somewhat less than in the primary market because it is cumbersome to unwind existing positions. To make the secondary market more liquid, several people have proposed the creation of a clearing corporation similar to the clearing corporations for futures and options. If this happens, the disadvantages for end-users would be less customization and more regulation. The advantages would be reduction in default (credit) risk and increased transparency. 


Market Conventions and Terminology 

The market convention for pricing swaps is to quote the fixed rate in terms of a basis point spread over the Treasury rate (usually quoted on a semi-annual bond-equivalent yield basis) as the price for receiving the floating-interest rate index flat (no basis points are added to or subtracted from the floating rate). For example, if an investor wants to receive a floating rate, such as LIBOR, the fixed rate it will have to pay would be the current on-the-run Treasury yield for the appropriate maturity category of the swap, plus a basis point spread over that yield (on-the-runs are the securities of the relevant maturity that were most recently auctioned). This basis point spread over the relevant Treasury is called the swap spread. For example, assuming that the on-the-run two-year Treasury yield is 6.00 percent and a two-year swap is quoted at 18/20 (bid/offer), then a fixed-rate receiver would pay the dealer LIBOR and receive a fixed rate of 6.18 percent, and a fixed-rate payer would pay the dealer 6.20 percent to receive LIBOR flat. 

It is important to distinguish between the swap spread and the bid/offer spread (discussed above in the primary market information). The swap spread is the spread over the Treasury yield to pay or receive fixed while the bid/offer spread is the difference between the fixed rate which must be paid to the market maker and the fixed rate that the market maker will pay. The swap spread represents the difference between investment-grade spreads (from Eurodollar futures and corporate bond markets) and the risk-free rate of Treasury securities. This spread adjustment is appropriate because non-U.S.-government swap counterparties typically cannot borrow at risk-free Treasury rates. The supply and demand for fixed-rate funds also influences the swap spread. For instance, if there is a predominance of fixed-rate payers in the market, swap spreads will increase as the demand for paying fixed on swaps will exceed the supply of dealers willing to book these swaps, thus bidding up the spread. 

Swaps are priced relative to other funding and investment vehicles with the same type of exposure. For shorter maturities, in which liquid interest-rate futures contracts are available, swaps are priced relative to futures contracts. Swaps of one- to five-year maturities are generally priced relative to Eurodollar futures. 

At longer maturities, swaps are priced relative to rates in alternative traditional fixed- and floating-rate instruments. For instance, swap spreads for 5- to 10-year maturities are roughly equivalent to investment-grade (single A or higher) corporate spreads over U.S. Treasuries .

Pricing Using Eurodollar Futures Contracts 

An interest-rate swap can be thought of as a series of forward contracts. As such, if forward rates are observable, a swap can be priced as a series of these forward contracts. Eurodollar futures contracts are observable, liquid market forward rates for U.S. dollar LIBOR. As the fixed rate on a swap is simply the blended forward rates for each floating reset date, swaps can be priced by reference to the Eurodollar strip (a series of Eurodollar futures contracts) out to the maturity date of the swaps contract. For example, consider a hypothetical one-year swap starting March 19, 1997, and terminating March 18, 1998 (March to March contract dates). 

Step 1: Determine forward rates by reference to the one-year Eurodollar strip.

Step 2: Calculate the swap rate based on the following formula:

R = ([1 + R0(D0 / 360)]
[1 + F1(D1/ 360)]
. . .
[1 + Fn(Dn/ 360)] - 1)

R = Eurodollar strip rate (swap rate) stated as an annualized money market yield 
R0 = spot LIBOR to first futures expiration
F1 = first futures contract (100 - futures price)
Fn = futures rate for the last relevant contract in the strip 
Di = actual number of days in each period

R = ([1 + .0575(91/ 360)]
[1 + .0593(91/ 360)]
[1 + .0618(91/ 360)]
[1 + .064(91/ 360)] - 1)
R = 6.21%

The above example is simplified because the swap begins and terminates on contract expiration dates. However, a similar methodology incorporating stub periods can be used to price swaps which do not fall on contract expiration dates by using the following generalized formula:

[1 + R0 (D0 / 360)]
[1 + F1(D1/ 360)]
. . .
[1 + Fn(D/ 360)]
= [1 + R(365/ 360)]N
[1 + R(Dr/ 360)]

Dr = total number of days in the partial-year period of the strip
N = number of whole years in the strip

Swaps are often priced using the Eurodollar strip for maturities of five years or less when liquidity in the Eurodollar strip is high. 

Pricing Using Zero-Coupon Methodology 

A zero-coupon methodology, another method used to value swap contracts, is often used to value swaps with maturities greater than five years. Unlike a yield-to-maturity (YTM) method in which each cash flow is valued at a constant discount rate, a zero-coupon methodology discounts each cash flow by a unique zero-coupon (spot) rate. A zero-coupon rate (zero) can be thought of as the YTM of a zero-coupon bond. As such, the return in period n on a zero-coupon bond can be derived by making n period investments at the current forward rates. For instance, the discount factors for a three-period instrument priced on a YTM basis would be derived as follows. 

YTM discount factors:

where YTM = constant yield-to-maturity rate.

The discount factors for a three-period instrument
priced on a zero-coupon basis would be derived as follows.

Zero-coupon discount factors:

Zero-coupon swap rates can be calculated either from the price of an appropriate zerocoupon swap or from a series of forward rates such as the Eurodollar futures strip. The market in zero-coupon swaps, however, is not active and zero-coupon prices are not observable. However, zero-coupon swap rates can be derived from observable coupon-bearing swaps trading in the market using a technique called bootstrapping. Once zero-coupon swap rates have been derived, an interest-rate swap can be priced similar to a fixed-rate bond by solving for the swap rate which, when discounted by the appropriate zero-coupon rates, will equate the swap to par. 

The first step in the bootstrapping method is to construct a swap yield curve based on coupon paying swaps trading in the market. Once this yield curve has been constructed, the coupon rates on the swaps can be used to calculate zero swap rates. Based on the observable first-period swap rate, a zero rate can be derived for the first period. Often, this rate may already be stated on a zero-coupon basis, such as six-month LIBOR (coupons are not paid on the instrument). The first period zero rate (z 1) is derived by discounting discounting the coupon rate on the first-period instrument by the zero-coupon rate which gives a price equal to par.


The first-period zero rate and the second-period coupon swap rate are then used to calculate the second-period zero rate (z2) using the following relationship:


This process is then continued to calculate an entire zero-rate curve. Zero rates for all other dates can then be calculated by interpolation. 

As an example of the zero-coupon pricing methodology, consider the following simplified example for a $100 million two-year amortizing fixed-for-floating interest-rate swap, quoted on an annual basis. The swap amortized by $50 million at the end of year one, and amortizes to zero at the end of year two. 

Step 1: Construct the cash-swap yield curve for two years.

Step 2: Derive the zero-coupon rates by the bootstrap method. 

Using the coupon swap rates from the swap yield curve above, the first-period zero-coupon rate can be solved using the bootstrap method:

Likewise, using the above cash-market swap rates to solve for the zero rate in year 2 by the bootstrap method:

Step 3: Using iteration, solve for the swap coupon rate which equates the cash flows on the swap to par using the zero rates obtained in step 2 as the discount factors.

$100mm = [$50mm+ (100mm * Swap Coupon)/(1.05)]+  [$50mm+ (50mm * Swap Coupon)/ (1.0602)2]
Swap-Coupon Rate = 5.65%

Pricing Unwinds

After a swap has been entered into, the mark-to-market (MTM) value can be calculated by discounting the remaining cash flows on the swap by the appropriate zero-coupon rates prevailing at the time of the termination of the swap. The resulting value, above or below par, would then represent the amount which would be either received or paid to terminate the swap. For example, using the amortizing swap example above, suppose that after one year, the counterparty who is a fixed-rate payer in the swap wishes to terminate the swap. At the time, one-year swap rates are 7.00 percent. The mark-to-market value of the swap would be calculated as follows:

Step 1: Determine the one-year (time remaining to maturity) zero-coupon rate.

Step 2: Discount remaining cash ows on the swap by the zero rate obtained in step 1.

Price of Swap = [$50mm + ($50mm .0565)] / (1.07)
Price of Swap = $49.37 mm
MTM Value = $50 mm - $49.37 mm = $630,000

In this example, as rates have risen since the inception of the swap, the fixed-rate payer would receive a fee of $630,000 for terminating the swap. 


Any firm that has a position in swaps is exposed to interest-rate, basis, and credit risks (discussed below). From a dealer standpoint, these risks are ideally hedged by entering immediately into mirror (offsetting) swaps, which eliminate exposure to these risks. However, in practice, dealers warehouse swap positions and hedge residual exposure with Eurodollar futures, forward rate agreements, or Treasuries until offsetting swaps can be established. End-users who have a swaps book face the same risks, and apply the same techniques, as dealers. 

Hedging Interest-Rate and Basis Risk 

Interest-rate risk in a swap portfolio is the risk that an adverse change in interest rates will cause the value of the portfolio to decline. Basis risk arises from an imperfect correlation between the hedge instrument and the instrument being hedged. Interest-rate and basis risk can be hedged one swap at a time (micro-hedging), or portfolio (set) of swaps can be hedged (macro-hedging). Micro-hedging is rare today. In macro-hedging, the overall risks of the portfolio (or subsets of it) are evaluated and hedged using offsetting interest-rate swaps and other interest-rate derivatives. Residual exposures are hedged in the Eurodollar futures or Treasury markets. Most dealers dynamically hedge the residual exposure of their swap portfolio by adjusting the hedge position as interest rates change. 

Risk managers usually take into account the effect of various interest-rate changes on the profitability of a swap book - for example, when interest rates change by 5, 10, 50, or 100 basis points. Dealers usually hedge for an arbitrary movement in rates, such as 50 basis points, which generally depends on senior management's risk appetite. 

Hedging Credit Risk 

The main techniques by which credit risk is hedged are (1) to require collateral if a counterparty is out of money; (2) to establish termination clauses in the master agreement for assessment of damages in the event of default; (3) to net payments (when several swaps are outstanding with the same counterparty), according to terms established in a master netting agreement (or master agreement); and (4) to sell the swap to another party. 

Hedging the credit risk of a swap book is difficult for a number of reasons. First, since there is no formal secondary market in swaps, it may not be immediately possible to trade out of a position. Second, assumptions about the certainty of cash flows and the level and term structure of interest rates are implicit in swap valuation. If these assumptions do not hold, the value of a swap book may not behave as expected, depending on how it is hedged. Third, to the extent to which some contracts are customized, they may be difficult to value accurately and to hedge. 

If risk models are used to estimate a market maker's potential future credit exposure, the assumptions between the risk-management model and the credit-risk model should be consistent. As is the case for risk management, it is important to understand the assumptions in the model in order to estimate potential credit risk. 


The principal risks in swap contracts are interest-rate, basis, credit, and legal and operating risk. For participants entering into highly customized transactions, liquidity risk may be important because hedging or an assignment of the contract may be difficult. 

Interest-Rate Risk 

Interest-rate risk for swaps is the risk that an adverse change in interest rates causes the swap's market value to decline. The price risk of interest-rate swaps is analogous to that of bonds. In fact, a swap can be described as an exchange of two securities: a hypothetical fixed-rate bond and a floating-rate note. The swap involves the simultaneous exchange of these two securities of equal amount and maturity, in which netting of principal payments at origination and maturity results in no principal cash flow. Along these lines, a swap dealer who makes fixed-rate payments is considered to be short the bond market. This dealer has established the price sensitivities of a longer-term liability and a floating-rate asset. The price risk here is that if short-term interest rates decrease, the dealer would be receiving less on the asset but still paying out the same amount on the liability. This interest-rate exposure could be hedged by buying Eurodollar futures (or by being long Treasuries of the same maturity as the swap). Then, if short-term interest rates decrease, the gain on the hedge should offset the loss on the swap. 

Basis Risk 

A major form of market risk that dealers are exposed to is basis risk. Dealers have to hedge the price exposure of swaps they write until offsetting swaps are entered into, and the hedges may not be perfect. 

Basis risk affects profitability. The bid/offer spread is the profit a dealer can make on a hedged swap book, but the dealer can earn less than this due to basis risk. 

Sources of Basis Risk 

When a dealer hedges swaps that have some credit risk with instruments of little or no credit risk (Treasuries), it creates basis risk. For instance, dealers often hedge swaps with maturities of five or more years with Treasuries. The risks in the swaps usually include credit risks, which are reflected in the floating rate(s). Since Treasuries are credit-risk-free securities, they do not provide a perfect hedge; this is a source of basis risk for the dealer, since there can be divergence between the two rates. Dealers are exposed to TED (Treasury-Eurodollar) spread risk when they hedge swaps of shorter maturities with Treasuries. In essence, the price of Eurodollar futures can change, which will cause swap spreads to change even if Treasury prices remain the same, since the swap spread is linked to the difference between the Eurodollar and Treasury markets. 

Credit Risk 

After the swap is executed, changes in interest rates cause the swap to move in the money for one counterparty and out of the money for the other. For example, an increase in market interest rates would increase the floating-rate payments from a swap, causing the value of the swap to the fixed-rate payer to rise, and the value of the swap to the floating-rate payer to fall. 

As no principal amount is exchanged in an interest-rate swap contract, credit risk is significantly less than it is on instruments in which principal is at risk. Credit-related loss can occur when the counterparty of an in-the-money swap defaults. The credit loss would be limited to the present value of the difference between the original and current market rates over the remaining maturity of the contract, which is called the replacement cost of the swap. For example, if a dealer had originally swapped fixed payments at 8.5 percent for six-month LIBOR for seven years, and the current market rate for the same transaction is 10 percent, the actual loss when a counterparty defaulted at the end of the first year would be the present value of 1.5 percent over six years on the notional principal amount of the swap. 

Credit risk is a function of both current credit exposure and potential future credit exposure. The example above only illustrates current credit exposure. Potential future exposure depends primarily on the volatility of interest rates. One approach to estimating peak potential credit exposure (PkCE) is to perform a full-blown Monte Carlo simulation on a counterparty's portfolio. This strategy has many appealing features and is the most statistically rigorous. In essence, the model is calculating "maximum" potential market value of the transaction, given a set of market conditions and a set confidence interval. However, problems arise from having to assume desired correlations among variables when making multiple simulations of market conditions. These correlations need to hold true over the life of the contract and be adjusted for the introduction of new instruments. Aside from these methodology problems, it is almost impossible to run the necessary number of simulated portfolio market values within response times acceptable to the trading floor. Also, Monte Carlo simulations do not readily highlight the specific sources of potential exposure or suggest ways to neutralize this exposure. 

An alternative to the full-blown Monte Carlo strategy can be characterized as the "primary risk-source approach.  "This approach attempts to identify the market variable that is the primary source of changes in the contractors value and then simulate values based on changes in this variable. In practice, a single market variable is not usually the only factor that causes a contractors value to change. However, other factors that might affect the value are generally of secondary importance. In addition, if the secondary-market variables are not highly correlated with the primary risk source, their impact on market value is further reduced. 

Estimating PkCE for a single contract can be complex. Accurately estimating PkCE for a portfolio of contracts executed with one counterparty can be so analytically difficult or computationally intensive that it is not always feasible. A trade-off has to be made between the ideal methodology and the computational demands. 

Other factors which affect potential credit exposure include the shape and level of the yield curve, the frequency of payments, the maturity of the transaction, and whether collateral has been posted. In addition, the changing credit quality of counterparties can affect potential credit risk. 

Legal Risk 

Legal risk arises from the possibility that a swap contract will not be enforceable or legally binding on the counterparty. For instance, the enforcement of netting agreements with foreign counterparties varies by country and may expose a counterparty to risk in case of non-enforceability. As such, the adequacy of legal documentation, including master swap agreements and netting agreements, should be reviewed. 


The accounting treatment for swap instruments is determined by the Financial Accounting Standards Board's Statement of Financial Accounting Standards (SFAS) No. 133, " Accounting for Derivatives and Hedging Activities." (See section 2120.1, "Accounting," for further discussion.) 


The credit-equivalent amount of an interest-rate swap contract is calculated by summing -

 1. the mark-to-market value (positive values only) of the contract and 
2. an estimate of the potential future credit exposure over the remaining life of each contract. 

The conversion factors are as follows. 

Remaining Maturity                   Factor 
One year or less                          0.00% 
Five years or less                        0.50% 
Greater than ve years               1.50% 

If a bank has multiple contracts with a counterparty and a qualifying bilateral contract with the counterparty, the bank may establish its current and potential credit exposures as net credit exposures. (See section 2110.1, "Capital Adequacy.") 


Swaps are not considered investments under 12 USC 24 (seventh). The use of these instruments is considered to be an activity incidental to banking within safe and sound banking practices. 


Arditti, Fred D. Derivatives. Harvard Business School Press, 1996. 
Beidelman, Carl R., ed. Interest Rate Swaps. Homewood, Ill.: Dow Jones-Irwin, 1990. 
Das, Satyajit. Swap and Derivative Financing. Chicago: Probus Publishing, 1994. Fabozzi, Frank J., ed. The Handbook of Fixed Income Securities. 2d ed. Homewood, Ill.: Dow Jones-Irwin, 1987. 
Global Derivatives Study Group. Derivatives: Practices and Principles. The Group of Thirty, July 1994. 
Hull, John C. Options, Futures and Other Derivative Securities. 2d ed. Prentice-Hall, 1993. 
Marshall, John F., and Kenneth R. Kapner. The Swaps Market. 2d ed. Miami, Fla.: Kolb Publishing, 1993. 
Smithson, C., C. Smith, and S. Wilford. Managing Financial Risk: A Guide to Derivative Products, Financial Engineering and Value Maximization. Richard D. Irwin, 1995. 4325.

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