# Default risk and innovations in the design of interest rate swaps.

In the past ten years, interest rate swaps have become one of the
most popular and effective products available to financial managers for
controlling the impact of changing financial market conditions. That
they are widely used is indisputable. Abken [1], citing data obtained
from the International Swap Dealers Association (ISDA), noted that by
year-end 1989 the U.S. dollar portion of the swap market (as measured by
outstanding notional principal) had grown to about $2.4 trillion on a
worldwide basis. This expansion is truly remarkable when one realizes
that the first single-currency interest rate swap was not transacted
until 1982. Further, the pervasiveness of these commitments is equally
impressive. For example, a recent survey of chief financial officers by
Institutional Investor [5] showed that nearly three-quarters of the
corporations with revenues of at least $3 billion have used swaps.

The explosive growth in the swap market over the past decade has masked one vital concern about these agreements. Simply put, though it is generally regarded as a tool for reducing interest rate risk, an interest rate swap is itself a risky transaction. Aggrawal [2] summarized the various types of exposure borne by the swap user, including risk due to unexpected changes in market prices and regulatory, legal, and accounting matters. But the most important of these risks is the potential economic loss that a firm would incur if its counterparty defaulted on the swap agreement when interest rates have moved adversely. Financial managers have now become aware of the default costs associated with using interest rate swaps, which, according to Cooper and Mello [6], are likely to be subordinate to debt claims in the event of bankruptcy. As an indication of this, the same Institutional Investor survey reported that roughly 80% of the CFOs responding expressed increasing concern over the creditworthiness of their swap partners, with over one-third of that group admitting to having actually rejected a potential counterparty because of this risk.

It is difficult to get accurate data on default losses in the swap market because many transactions apparently are worked out without technical default. Kapner and Marshall [9] reported results from a 1987-1988 ISDA survey indicating that only 11 of 71 swap dealers had experienced write-offs, amounting to just $33 million on aggregated portfolios at that time of $283 billion. While this figure suggests very low default rates, it pre- dates the court-ordered default on swap positions held by British local authorities (i.e., municipalities) whereby a number of bank counterparties stand to lose amounts estimated at (pounds)500 million on positions that reached a maximum notional principal of over [pounds]10 billion. In that case, the courts ruled that the local authorities had exceeded their legal power because the swap positions could not be justified as hedges and thus were considered outright speculation. Further, although a 1992 ISDA survey indicated that only about 0.5% of the outstanding notional principal over the last decade was lost to default, Glasgall and Javetski [7] noted that workouts were also needed to transfer positions involving the failed institutions, Bank of New England and Drexel Burnham Lambert, with the latter's swap book totaling over $30 billion.

There are two ways of viewing swap default risk: (i) actual exposure, which is a measure of the loss if the counterparty were to default and is based on the movement in swap market rates between the inception of the agreement and the current date; and (ii) potential exposure, which is based on a forecast of how market conditions might change between the present and the swap's maturity date, including in some manner the probability of default by the counterparty. Thus far, the academic literature investigating this topic has concentrated on estimating the value of potential exposure and showing how it is shared between the counterparties. Simons [11] demonstrated how many of the ad hoc estimation techniques used in practice could be improved by employing simulations of different future interest rate environments. This approach was generalized by Sundaresan [13] who assumed a stochastic process to characterize future interest rate movements in order to show how default risk impacts swap valuation. Cooper and Mello [6], in addition to producing a theoretical model for valuing potential default risk, examined the transfer of this risk between three groups associated with a firm: shareholders, debtholders, and the swap counterparty. While this focus on potential exposure is important, what remains largely unaddressed is the extent of the actual default risk that accumulates as a swap position seasons. The goal of the present research is to explore ways in which the structure of the swap agreement can be revised to reduce this type of exposure.

The analysis is organized as follows. In the first section, we develop an analytic framework to measure actual default exposure and discuss the market practices widely used to deal with that risk. Section II outlines the mechanics of the mark-to-market swap, the first of two design innovations we examine. We demonstrate that the mark-to-market swap agreement is essentially a sequence of swap commitments, each of which is liquidated after a single settlement period. A second design innovation, the forward rate swap, is introduced in Section III. In this approach, the usual swap structure is modified at the time of its origination by altering the pattern of fixed swap rates to reflect expectations about the market conditions that will prevail on the future settlement dates. Examples of both designs, along with a discussion of practical issues in their implementation, are presented in each respective section. In the final section, we offer some concluding comments.

I. Default Risk on an Interest Rate Swap

A. Measuring Default Risk Exposure

Default risk exposure on an interest rate swap differs from that on an ordinary bond in three important ways. First, the most common (i.e., "plain vanilla") swap structure requires no exchange of principal payments at inception or maturity, establishing instead a notional principal used to convert interest rates into cash flows for each settlement period. Therefore, there is no principal at risk. Second, a swap is an executory contract, signifying that one party need perform only if the counterparty also performs. If one party ceases paying the fixed rate, the other need not pay the floating rate. Combined, these two considerations indicate why the risk exposure on a swap is significantly less than that on a bond of comparable face value. In fact, the exposure is limited to the difference between the present values of the remaining fixed-rate and floating-rate cash flows. Because that net present value may be positive or negative, the third important difference between a swap and a bond is that the default risk exposure is bilateral in that each party must be concerned about the possible default of the other.

To measure default risk, suppose that on date-T (i.e., T periods ago, relative to the current date 0) Firms A and B agreed to a swap transaction wherein Firm A makes payments based on a prespecified fixed rate every settlement date in exchange for receipt of cash flows based on a variable interest rate index. Settlement typically is on a net basis, the difference between the fixed and floating flows. This transaction, assumed to mature at date N, will then have a total of T + N settlement dates over its lifetime. Let [F.sub.-T, N + T] be the fixed rate in the transaction (denoting the rate established at date -T for a swap with T + N periods to maturity) and [I.sub.t-1] be the index rate observed at date t - 1 and payable at date t. Swaps typically settle in arrears in that the floating rate is set at the beginning of the period and payment is made at the end.

Now suppose that immediately after the settlement payment (or receipt) on date 0, Firm B enters bankruptcy proceedings, thereby defaulting on the remaining N settlement obligations. Assume that on date 0 the current market fixed rate on a swap with N periods to maturity is [F.sub.0, N], a rate assumed to be higher than [F.sub.-T, N + T]. As the payer of what had been a "below- market" fixed rate, Firm A experiences a loss equal to the following amount:

[Mathematical Expression Omitted]

where [V.sub.0] is the economic value of the swap on date 0, [r.sub.t] is the (zero-coupon) risk-adjusted discount rate appropriate for each of the remaining N cash flows, and NP is the notional principal on the swap. The value of the swap to Firm A on date 0 is the discounted value of the difference between the cash flows generated by the higher fixed rate on a replacement swap ([F.sub.0, N]) and its original level ([F.sub.-T, N + T]). If [V.sub.0] were negative, implying that [F.sub.0, N] is less than [F.sub.-T, N + T], Equation (1) would represent the amount that Firm A would gain if its counterparty were to default. In that case, Firm B would be receiving an above- market fixed rate for payment of the floating-rate index; the swap would represent a valuable asset rather than a liability to be extinguished in bankruptcy. Clearly, Firm B would choose not to default on the agreement, rather it would prefer to continue the transaction or attempt to transfer it to a third party at a profit.

Firm A's actual risk exposure (RE) following default by Firm B at date 0 can then be summarized by:

[Mathematical Expression Omitted]

Similarly, as the fixed-rate receiver, Firm B's risk exposure at date 0 is:

[Mathematical Expression Omitted]

Equations (2) and (3) are the mark-to-market risk exposures for Firms A and B in that they summarize the loss that would be incurred based on the current swap market rate. Notice that the exposures depend on the joint occurrence of two events: (i) the default by the counterparty, and (ii) an adverse movement in interest rates. While potential default risk at origination may be bilateral, because of this second condition the actual exposures at any subsequent date will be unilateral. That is, it will never be the case that both parties simultaneously have positive mark-to-market exposures to one another.(1)

How default risk exposure evolves over time depends fundamentally on the replacement swap fixed rate, [F.sub.0, N], and the time remaining in the contract, N. Generally, the more that [F.sub.0, N] diverges from the existing rate, [F.sub.-T, N+T], and the larger is N, the greater is the default loss. Early in the life of the swap, when N is high but [F.sub.0, N] is likely to be relatively close to [F.sub.-T, N+T], the exposure would be relatively small. At an extreme, if the counterparty enters bankruptcy proceedings immediately after signing the swap document, the default loss is likely to be negligible given that the firm presumably can replace the swap at the same market rate. Late in the life of the swap when N is low, the exposure again can be small despite a significant spread between [F.sub.0, N] and [F.sub.-T, N+T]. In fact, at maturity, the risk exposure cannot exceed the difference between the floating-rate index at the last settlement date ([I.sub.N-1]) and the fixed rate, times the notional principal. Overall, mark-to-market risk exposure is likely to reach an internal maximum at some time between the origination and maturity dates.

B. Market Practices to Deal With Default Risk

Most swap market makers are either commercial or investment banks. Because of the fundamental difference in the nature of their business, however, they deal with default risk on swaps differently. Commercial banks are in the credit-risk-bearing business -- that is their chief function and source of market expertise.(2) As such, they have tended to price default risk explicitly into the fixed rate on the swap, raising their receive-fixed (offer) rate and lowering their pay-fixed (bid) rate as compensation to the extent that the counterparty represents additional risk. As would be expected, the presence of greater risk widens the market maker's bid/offer spread. Similarly, entering a swap with a stronger counterparty would induce the bank to narrow its bid/offer spread. Investment banks, on the other hand, are not in the business of bearing the credit risk of customers over the long-term. Consequently, they have tended to use letters of credit and posting of collateral to raise the effective credit standing of weaker counterparties to their level. Rather than pricing the default risk of a particular firm into the fixed rate on the swap, investment banks would have a uniform bid/offer spread applicable only to acceptable counterparties.

The key point is that it is the imbalance in degrees of credit risk between the counterparties that motivates the adjustment in the fixed rate on the swap or the request for collateral (or letter of credit). Given that each counterparty must assess its default risk exposure to the other, neither would be very willing to give up something in the agreement without getting the same in return. So, collateral is used in the swap market mostly to equalize a disparity in the creditworthiness of the counterparties, not to manage the inevitable exposure that arises even between two comparable firms.(3) Mutual exchange of collateral presents the problem of retrieving the securities that had been posted if the counterparty were to default on the swap, especially if those securities had been lent out in the "repo" market. An alternative would be to post collateral with a third party acting as a clearinghouse. The value of securities to be posted could be a fraction of the notional principal or the market value of swap. In any case, the use of collateral raises transactions costs in the same manner as the margin account on an exchange-traded futures contract.

Because of the bilateral nature of potential credit risk, swaps are negotiated on a more "level playing field" than typically encountered with bank loan contracts. That, in turn, also limits the ability of both parties to demand restrictive covenants. However, there are two such items that usually are contained in swap documentation. The first is a cross-default clause which triggers the swap's default in the event that any indebtedness of one of the counterparties is either in default or has been accelerated. This provision mitigates the effects of what we describe below as a default-timing option. Second, swap agreements also often require that multiple contracts between the same counterparties be netted against each other, which eliminates the ability of the defaulting party to "cherry-pick" contracts -- i.e., default on only those with negative economic value while maintaining the ones with positive value. To guarantee that provisions such as these are adopted, market makers prefer that customers sign a master swap agreement to expedite future transactions and ensure that all existing swap positions can be netted out if a default were to occur.

A default-timing option can arise if the defaulting party can benefit by postponing the event of default itself. Suppose that the economic value of the swap, as measured by Equation (1), is negative but the firm is scheduled to receive a payment at the next settlement date. For example, assume Firm B is in serious financial distress at a time when [F.sub.0, N] [is greater than] [F.sub.-T, N+T] and [I.sub.0] [is less than] [F.sub.-T, N+T]. This means that the existing receive-fixed rate is below-market even though the next scheduled settlement provides a net cash inflow. This could occur if the yield curve dramatically steepened after the swap was originated, driving the short-term index rate down and the longer-term fixed rate up. Firm B clearly would have no reason to default on the swap prior to the next receipt. In general, it would have an incentive to defer defaulting on the swap as long as the floating-rate index calls for net cash inflows. Moreover, all creditors of Firm B would concur with the decision to defer default until a time when [F.sub.0, N] [is greater than] [F.sub.-T, N+T] and [I.sub.0] [is greater than] [F.sub.-T, N+T].(4) As noted, a cross-default clause works against such a default-timing option by identifying as events of default factors other than direct nonpayment of an owed amount on a settlement date; for instance, filing of bankruptcy proceedings by any other creditor will also bring the swap into default, triggering termination of the agreement.

II. Swap Design Innovations: Mark-to-Market Swaps

A. Revising the Structure of the Swap Agreement

We have shown that any firm engaged in an interest rate swap will incur a loss if its counterparty defaults at a time when interest rates have moved in an adverse direction. One way to eliminate this exposure is to have the participants liquidate their existing agreement every time interest rates change and then renegotiate the terms of a new swap to reflect the prevailing market conditions. Of course, because interest rates are never static this solution is impractical in that it would require virtually perpetual exchanges of mark-to-market cash flows. As a more manageable alternative, the counterparties could agree to a scheme in which the renegotiations occur only on the periodic settlement dates of the swap (i.e., when cash flow exchanges were scheduled to occur anyway). That is, imagine that Firms A and B had structured the following agreement at date-T:

(i) Enter into a plain vanilla swap for T + N periods with Firm A paying, and Firm B receiving, the fixed rate of [F.sub.-T, N+T];

(ii) On the first settlement date (i.e., date -T + 1), the counterparties: (a) make their respective fixed- and floating-rate payments on the existing swap; and (b) liquidate the remaining portion of the existing swap according to Equation (1) above, using the current swap rate as the discount factor for all future cash flows, i.e., set [r.sub.t] equal to [F.sub.-T+1, N+T-1], which would be the fixed rate as of date -T + 1 on swap having N + T - 1 periods until maturity;

(iii) Immediately upon unwinding the old swap, the participants enter into a new swap having the same notional principal as the original but (N + T - 1) periods to maturity. The new fixed rate on this swap, again, would be [F.sub.-T+1, N+T-1];

(iv) Repeat steps (ii) and (iii) on each settlement date until the maturity of the original swap (i.e., date N).

This procedure, though admittedly contrived, captures the essence of the mark-to-market swap transaction. Formally, rather than having the counterparties enter and then unwind (N + T - 1) different swaps -- as is implied by the above scheme -- the mark-to-market swap would be a single agreement made at date -T and terminating at date N. The difference between it and the plain vanilla commitment covering the same period is that while the latter generates default risk exposure between dates -T and 0 by leaving the fixed rate set at [F.sub.-T, N+T], the mark-to-market swap eliminates this exposure on each settlement period. It does this by having the counterparties settle in cash any change in the swap's economic value and then resetting the fixed rate on the remainder of the agreement to the prevailing level. Thus, Firm A's settlement payment at date 0 can be written:

[Mathematical Expression Omitted]

where, as noted above, the new swap fixed rate is used as the common discount factor for all future cash flows. Notice that Equation (4) consists of two parts, the first being the scheduled net settlement payment on the existing swap and the second being the present value of unwinding the remainder of that swap according to Equation (1).

Beyond its ability to reduce default risk, another key property of this "marking" procedure is that the overall internal rate of return on a swap-linked variable rate funding structure is relatively invariant to the exact swap rate that prevails on each future settlement date. This follows directly from the trade-off that is created between making (receiving) a swap unwind payment and then adjusting the remainder of the agreement to a lower (higher) rate. Consequently, no matter what path interest rates follow in the future, the swap unwind payment -- and subsequent resetting of the fixed rate on the "new" swap -- always link the process back to the conditions in the market at date 0. This point is illustrated by the following example and documented more thoroughly in the Appendix.

B. A Numerical Example

Suppose that a corporation has just issued a two-year, floating-rate note (FRN) with a par value of $10 million, calling for semiannual interest payments based on the London Interbank Offer Rate (LIBOR). Because this firm presumably would prefer a known cost of funds but now would be exposed to rising rates, it can convert its new debt into the equivalent of a fixed-rate issue via the swap market. This can be accomplished by entering a pay-fixed, two-year (i.e., four settlement period) swap against six-month LIBOR with a notional principal of $10 million. That is, as floating-rate payments are required on its FRN, the firm will receive an equal amount from its swap counterparty in exchange for cash flows based on the fixed swap rate. Letting the current fixed rate for such an agreement be [F.sub.0, 4] = 9.00%, Exhibit 2 summarizes the data necessary to illustrate the mechanics of the procedure outlined above.(5)

Panel A specifies one possible pattern for swap fixed rates on each future settlement date. Note that with each successive settlement date, the maturity of the swap is reduced by six months to match the remaining maturity on the underlying note. Panel B shows the calculations of the periodic cash flows associated with these assumptions, including both the fixed payments on the existing swap and the liquidation value of the remainder of that swap.(6) For example, on date 1 the economic value of the existing agreement is [V.sub.0] = -$69,049, a negative amount since the firm is obligated to pay 9.00% when the current market rate is only [F.sub.1, 3] = 8.50%. The corporation would have to pay its counterparty that amount to close out the existing agreement. In addition, the fixed payment due on that date is $450,000, given the initial fixed rate of 9.00%. Finally, Panel C computes the corporation's synthetic funding cost, expressed as the internal rate of return of the combined FRN/swap transaction.

This example highlights two important attributes of the mark-to-market swap. First, and most importantly, notice that when measured to the basis point the internal rate of return on the aggregate position is 9.00%. Of course, this is the same net funding rate that the firm would have had if it could have found a counterparty willing to accept 9.00% as a fixed rate on a plain vanilla swap with otherwise comparable terms. Thus, when viewed solely in terms of the cost of funds over the entire two years, the plain vanilla and mark-to-market swap structures achieve the same result. Second, unlike the plain vanilla swap, which would have required constant settlement payments of $450,000, the mark-to-market scheme generates highly variable cash flows from period to period. The exact payment pattern will depend on the path of the fixed rates on the sequence of replacement swaps, which in practice could come from the market maker's own quote sheet or as the median of a set of quotes from competing swap dealers. We should stress, though, that the plain vanilla swap keeps the periodic cash flows constant at the expense of creating default risk exposure for one counterparty or the other. The whole point of the mark-to-market structure is to eliminate this exposure on each settlement date by converting it to an obligatory cash payment.

C. Practical Considerations

Mark-to-market swaps have been traded in some form since 1987. Originally created by Manufacturers Hanover Trust Company as a means of giving less creditworthy firms access to the swap market, the product has gained slow, but steady, acceptance. Two difficulties often are cited by practitioners as barriers to implementing this innovation. First, many counterparties lack either the expertise or the back office technical support necessary to track the cash flow changes during the life of the agreement. In such cases, the only alternative is to rely exclusively on the calculations of the swap dealer, something many firms are understandably reluctant to do. This is a particular concern for a sizable portion of the professional (i.e., bank) counterparty market that has found it difficult to adapt existing accounting and computing systems to handle a large volume of anything other than plain vanilla transactions.

A second reason for the reluctance of many corporate counterparties to use the mark-to-market scheme stems from the unpredictability of the cash flow streams it creates. More precisely, this swap structure might obligate the counterparty to make sizable unwind payments prior to maturity. For instance, at date 1 in the above example, the firm had to pay almost $70,000 more than it expected based on information available at date 0. A company that actually has to raise this money in the capital market may find little consolation in the fact that its settlement payments will then be lowered for the remainder of the swap. This is a particular concern to industrial corporations who may not access their capital sources with the same frequency as do financial firms. Further, if the firm does have to tap external funding sources, its internal rate of return will only equal the original fixed rate on the plain vanilla agreement if its borrowing cost is equal to the prevailing fixed swap rate used as a discount factor. However, if the firm's cost of capital exceeds [F.sub.0, N], its aggregate funding cost from the bond/swap combination will exceed [F.sub.-T, N+T].(7)

On the other hand, participants in the swap market have found two features of the mark-to-market structure to be quite attractive. The most appealing attribute, of course, is the ability to mitigate the actual default risk exposure to their swap counterparties. In principle, it is possible to reduce this exposure even further by marking the agreement more frequently, perhaps using a system of margin accounts to collateralize the adjustment process. Practically, though, this variation creates measurable costs for the firm in that additional resources would have to be committed to monitoring its position. Further, it is likely to violate the guidelines created by the Commodity Futures Trading Commission to keep the swap and exchange-traded futures markets distinct. The second desirable feature of the mark-to-market format is that neither counterparty is hurt by a miscalculation of the new fixed swap rate on each settlement date. That is, if the reset rate agreed to on date t is ([F.sub.t, N-t] + [Epsilon]), as opposed to its "true" level of [F.sub.t, N-t], the fixed-rate payer would then be required to make higher future settlement payments but receive the present value of this amount as an unwind payment at date t. Again, this stresses that the pattern of future reset rates -- whether driven by market forces or human error -- is not by itself an important factor.

III. Swap Design Innovations: Forward Rate Swaps

A. Departures from the Plain Vanilla Swap Structure

The inherent structure of a plain vanilla interest rate swap, whereby a single fixed rate applies to all settlement dates, is itself a source of default risk. This is because in virtually all yield curve environments, there will be actual default risk exposure associated with a given contract even if interest rate movements are nonstochastic. In other words, if the future path for the floating-rate index were fully deterministic, a firm would typically incur a financial loss if its swap counterparty defaulted. The exceptional circumstance when this might not occur is an extended period of constant interest rates and a totally flat yield curve. That rarity aside, default risk arises because interest rates are stochastic, a reality exacerbated by the setting of a uniform fixed rate for the lifetime of the swap.

Consider again the fixed rate ([F.sub.0, N]) on a new N-period swap set on date 0. For both counterparties to the swap, the sequence of expected future payments and receipts is determined by the sign of [F.sub.0, N] - E([I.sub.t-1]). However, assuming risk-neutral participants, the swap market will be in equilibrium only if the present value of the fixed flows set by [F.sub.0, N] equal the present value of the expected variable flows set according to [I.sub.t]. This, in turn, means that the swap has an initial economic value of zero to each counterparty if the following condition holds:

[Mathematical Expression Omitted]

Note from Equation (5) that the initial exchange at the end of the first period is based on the known, current level of the floating- rate index ([I.sub.0]) while subsequent exchanges are only known to the level of their expected values.

Suppose that the overall market environment is one of expected rising short-term rates, reflected by an upward-sloping yield curve. This means that as the single swap fixed rate, [F.sub.0, N], will be a complex average of the yields expected to prevail during the N-period life of the agreement. Consequently, for the fixed- payer (fixed-receiver) on the swap, there is likely to be some future date t* after (before) which the firm expects to be receiving cash settlement payments and before (after) which it expects to be making payments. To the fixed-payer, the default risk exposure is then "back-loaded" in that it would be more concerned about its counterparty defaulting in the later years of the agreement when receipts are anticipated. For the fixed-receiver, on the other hand, the default risk would be front-loaded because it expects to be receiving settlement payments early and to be paying later. These conclusions, of course, implicitly rely on the expectations theory of the yield curve and the parties to the swap having subjective expectations aligned with the overall market.

A departure from the structure of a plain vanilla swap to address this front-loading and back-loading problem would be to set a time-varying fixed rate on the swap. That is, the fixed rate would be prespecified for all future settlement periods at inception but it would differ for each period so as to minimize expected settlement payments and receipts, thereby minimizing the expected default risk. For example, suppose that the uniform fixed rate is replaced in Equation (5) with the set of implied forward rates known at date O, based on the swap yield curve for maturities ranging from one to N periods.

[Mathematical Expression Omitted]

where IF[R.sub.0, 1] is the observed rate (e.g., LIBOR) for a transaction between dates 0 and 1 and IF[R.sub.t-1, t] is the implied forward rate on a one-period security between dates t -1 and t. An agreement structured in this fashion is what we call a forward rate swap. It is not a forward swap in the sense of a having a deferred starting date, rather it contains a sequence of forward fixed rates for future exchanges. This sequence need not be limited to just implied forward rates calculated from spot market quotations; it could come from observed rates in the explicit forward or futures market. For example, Eurodollar futures and over-the-counter forward rate agreement (FRA) markets could be used to establish the sequence.(8) In fact, the set of rates could simply be mutually acceptable to the two counter parties based on their own subjective probabilities of future rates.

The reduction in default risk exposure depends critically on the extent to which the actual path for the floating-rate index follows the trajectory of the forward rates. If the difference between the two is small, default risk can be substantially reduced. For instance, if IF[R.sub.t-1, t] is a reasonably good measure of E([I.sub.t-1]), the expected payoff for each future settlement date should be less than setting a single fixed rate. Note that the implied forward rate need not be a perfect predictor of future market rates for default risk exposure to be reduced as long as the variance of (IF[R.sub.t-1, t] - [I.sub.t-1]) is less than the variance of ([F.sub.0, N] - [I.sub.t-1]). However, it should be emphasized that if interest rates move significantly away from the implied forward path, the default risk can be even higher than on a plain vanilla swap structure. So, unlike the mark-to-market swap, a forward swap might not always reduce default risk.

B. A Numerical Example

Suppose that the fixed rates on one-, two-, and three-year annual settlement interest rate swaps (against twelve-month LIBOR) are [F.sub.0, 1] = 8%, [F.sub.0, 2] = 10%, and [F.sub.0, 3] = 11%, respectively. Current twelve-month LIBOR is also eight percent, so the one-year swap quote really only provides a benchmark -- it would be an agreement to both pay and receive the same known rate. Thus, if a firm enters the three-year swap as the fixed-rate payer at the market rate of 11%, it knows in advance that it is scheduled to make a settlement payment in one year equal to [F.sub.0, 3] - [I.sub.0] = 3% times the notional principal. Similarly, the fixed-rate receiver is assured of the receipt of the same amount, barring default by the counterparty. Clearly, the market levels for LIBOR in one and two years (i.e., [I.sub.1] and [I.sub.2]) will determine which counterparty ultimately receives the greater cash inflows. The fixed-rate payer gains if future LIBOR rises sufficiently above 11% in future periods to compensate for the first year's known cash outflow.

The implied forward rates based on this swap yield curve are IF[R.sub.0, 1] = 8% (= [I.sub.0]) for the first year, IF[R.sub.1, 2] = 12.245% for the second, and IF[R.sub.2, 3] = 13.408% for the third.(9) The forward rate swap structure would set the fixed rate for the first-year settlement at 8%, the second-year settlement at 12.245 %, and the third-year at 13.408%. That this swap has the same initial economic value as the plain vanilla structure using a single fixed rate of 11% for each period can be demonstrated by calculating the present values of each fixed cash flow stream:

8%/1.08 + 12.245%/[(1.10102).sup.2] + 13.408%/[(1.11193).sup.3] = 27.261%

= 11%/1.08 + 11%/[(1.10102).sup.2] + 11%/[(1.11193).sup.3]. (7)

While the plain vanilla and forward rate structures have the same initial economic value, the degrees of default risk exposure and the settlement cash flows will differ by a considerable amount. Suppose that a firm agrees to pay the fixed rate on a swap and that LIBOR does follow the implied forward rate trajectory (i.e., twelve-month LIBOR rises from [I.sub.0] = 8% to IF[R.sub.1, 2] = 12.245% on the next settlement date). With a plain vanilla swap, the firm would be obligated on that date to make a settlement payment of three percent times the notional principal because LIBOR at inception was eight percent. In addition to that payment, the swap would have positive mark-to-market value given that the two-year replacement swap fixed rate would be above 11%. With 12-month LIBOR equal to 12.245% on that date and an expected rate for the following year of 13.408%, the replacement swap fixed rate [F.sub.1, 2] can be calculated as follows:

12.245%/1.12245 + 13.408%/[(1.13408).sup.2] = [F.sub.1, 2]/1.12245 + [F.sub.1, 2]/[(1.13408).sup.2]. (8)

Solving for [F.sub.1, 2] obtains 12.726%. The economic value of the swap, and hence the default risk exposure, is the present value of the difference between 12.726% and 11% for the next two settlement dates, an amount equal to 2.875% (times the notional principal). That amount, by the way, is what would be received by the firm in a mark-to-market swap in exchange for resetting the fixed rate to 12.726% for the remaining settlement dates. In sharp contrast, a forward rate swap in the same environment would entail no settlement payment or receipt on that date. Moreover, the default risk exposure would be zero. It follows that this structure minimizes default risk to the extent that the actual trajectory for the floating-rate index follows the path set for the time-varying fixed rate on the swap. What matters is that actual rate movements are positively correlated with the scheduled changes in the fixed rate on the swap.

C. Practical Considerations

While there is an emerging market for mark-to-market swaps, we know of no actual swap transactions that employ the forward rate structure to mitigate default risk exposure. One possible reason for this could be similar back office technical difficulties as with mark-to-market swaps. A time-varying fixed rate would require some monitoring and investment in accounting and information systems, which, while demanding an increased time commitment from the company's treasury unit, would likely only burden the first-time user. The more plausible reason, however, is that swaps between commercial banks and their corporate customers have tended to have the bank receiving, and the corporation paying, the fixed rate. In addition, yield curves have been upward-sloping in recent years, especially in the intermediate-term maturities that typify swap agreements. These two circumstances combined have moderated concern over default risk from the perspective of the bank, but not necessarily so for the corporate end-user. On plain vanilla swap contracts, the corporations that are paying the fixed rates would be making net settlement payments in the early years of the transaction and receiving them in the later years. This is preferable for the bank, since it is much easier to assess the probability of financial distress in the near term than in the more distant future.

When the yield curve is upward-sloping, the proposed forward rate structure could even exacerbate the default risk exposure from the perspective of the commercial banks. In the later years of the swap, when the fixed rate would be higher than if a single rate applied to all periods, a default by a counterparty scheduled to be paying the fixed rate would be more costly to the bank. This suggests the three related conditions for the introduction of forward rate swaps: (i) corporations becoming more leery of banks as counterparties and seeking out swap designs that minimize default risk from their own perspective; (ii) an extended period of inverted yield curves, such as prevailed in the early 1980s in the United States and more recently in the United Kingdom; and (iii) more corporate demand to receive the fixed rate on interest rate swaps.

IV. Concluding Comments

The two design innovations we discuss, mark-to-market swaps and forward rate swaps, both draw the structure of the agreement closer to that of an exchange-traded futures contract. Futures exchanges manage their default risk by daily mark-to-market valuation and settlement transactions posted to a margin account. That this is done on a daily basis owes to the key role played by "locals" -- individual traders who provide much of the liquidity to the market but are thinly capitalized compared to institutional participants, commercial and investment banks, and major corporations. The fact that plain vanilla swaps do not typically require margin accounts with daily settlement (or, more generally, any form of collateralization) has been central in their appeal to corporate managers of financial risk. So these innovations, which would introduce periodic mark-to-market settlement and time-varying fixed rates, borrow from the futures market, yet still maintain a separate identity.

Default risk arises because firms do fall into financial distress and market interest rates are stochastic. The mark-to-market design for the swap uses actual rates to make ex post adjustments to the fixed rate. The forward rate design uses expected rates to make ex ante adjustments. Each design can effectively transform a floating-rate liability to a fixed-rate liability with a known, locked-in cost of funds subject, of course, to some assumptions about reinvestment rates. However, the amount of the net interest payment for each period is not known in advance with the mark-to-market design. With the forward rate swap, the future amount is known but is likely to differ for each period. The appeal of the plain vanilla structure is that it can provide a constant, known net interest payment for all periods. The trade-off is one of certainty with respect to future cash flows versus default risk. The mark-to-market swap manages default risk most effectively, limiting exposure to rate changes only over the most recent settlement period. The forward rate swap manages the risk less precisely and only to the extent that future rates follow the path structure in the agreement. Market rate changes accumulate on both the plain vanilla and forward rate structures, and can lead to a much larger amount of default risk exposure, given the direction and extent of the rate movements and the time remaining until maturity. Ultimately, application of these innovative designs for swap contracts will depend on economic conditions -- as risks arise, techniques and product designs always emerge to manage those risks.

1 In addition to mark-to-market exposure, which represents the actual consequence of a swap default at a specific point in time, many swap market participants are also concerned with estimating the maximum potential risk over the agreement's lifetime. These counterparties, typically money-center banks concerned about their total credit exposure to a corporate client, often determine a measure called fractional exposure. Although we have found that procedures for calculating fractional exposure vary across institutions, it is essentially an estimate of the worst-case scenario for mark-to-market risk between the present date and the swap's maturity. Thus, fractional exposure is analogous to the potential default risk measure analyzed by Simons [11] and Cooper and Mello [6].

2 A motivation for commercial banks to enter and promote the swap market has been to generate off-balance-sheet income and to exploit their comparative advantage in credit-based products at a time when traditional lending to corporate customers was falling off. Brown and Smith [3] described this as the "reintermediation" of commercial banking.

3 This follows, in part, from the idea advanced in Brown and Smith [4] that there will be a common fixed rate for all swap transactions involving counterparties of comparable creditworthiness. That is, in equilibrium, two AA-rated firms will negotiate to the same fixed rate as two BBB-rated credits.

4 There also is a plausible scenario whereby Firm B would elect not to default even though both [F.sub.0, N] [is greater than] [F.sub.- T, N+T] and [I.sub.0] [is greater than] [F.sub.-T, N+T]. Depending on the level of [I.sub.0], the firm could choose to make a small net settlement payment to keep the swap alive, in hope that swap market fixed rates subsequently fall and the swap takes on positive economic value. In effect, the periodic settlement payment is the premium for an out-of-the-money option on an interest rate swap. The value of that option might exceed its purchase price.

5 It should be noted that this example is typical of how corporations participate in the swap market. In fact, Wall and Pringle [14] provided survey data indicating that large companies are five times more likely to be fixed-rate payers than fixed-rate receivers. This same study also contains an excellent discussion of the many motivations for using swaps.

6 In these calculations, two simplifying adjustments have been made to Equation (4). First, the scheduled settlement payment is listed on a gross basis, rather than net of the LIBOR-based swap receipt. This was done because the floating-rate swap receipt will merely be passed through to the firm's bondholders and can therefore be ignored. Second, all interest rates were altered to reflect semiannual payment periods even though they were originally expressed on an annual basis.

7 A similar statement can be made for the unwind funds received on each settlement date. If these receipts cannot be reinvested at [F.sub.0, N], then the actual internal rate of return for the combined transaction will not be equal to [F.sub.-T, N+T].

8 This is consistent with the observation, expressed in Smith, Smithson, and Wakeman [12] and Kawaller [10] among other places, that plain vanilla interest rate swaps can be replicated to some degree by strips of either over-the-counter forward or exchange-traded futures contracts on the index interest rate.

9 To calculate the implied forward rates, one first needs to derive (or observe in the market) the zero-coupon rates that are consistent with the prevailing yield curve on coupon securities. In this context, the phrase "consistent with" means that there would be no arbitrage profits available from buying the coupon bond and selling its coupon and principal cash flows as separate zero-coupon instruments, nor from buying zeros and "reconstituting" the coupon bond. Iben [8] has provided a discussion of this technique and its application in the swap market. For example, the two-year zero-coupon rate of 10.102% is found as the solution for [Z.sub.2] in: 100 = 10/1.08 + 110/[(1 + [Z.sub.2]).sup.2]. The par value purchase price of the two-year bond must equal the present value of the future cash flows. Similarly, the three-year rate of 11.193% is found as the solution for [Z.sub.3] in: 100 = 11/1.08 + 11/[(1.10102).sup.2] + 111/[(1 + [Z.sub.3]).sup.3]. Notice that the first-year's coupon flow of 11 is discounted by the known eight percent one-year rate and the second year's flow is discounted by 10.102%, the result of the previous calculation. The implied forward rates are then calculated in the usual manner as: IF[R.sub.1, 2] = [[(1.10102).sup.2]/(1.08)] - 1 = 0.12245 and IF[R.sub.2, 3] = [[(1.11193).sup.3]/[(1.10102).sup.2]] - 1 = 0.13408.

References

1. P. Abken, "Beyond Plain Vanilla: A Taxonomy of Swaps," Federal Reserve Bank of Atlanta Economic Review (March/April 1991), pp. 12-29.

2. R. Aggrawal, "Assessing Default Risk in Interest Rate Swaps," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 430-448.

3. K. Brown and D. Smith, "Recent Innovations in Interest Rate Risk Management and the Reintermediation of Commercial Banking," Financial Management (Winter 1988), pp. 45-58.

4. K. Brown and D. Smith, "Plain Vanilla Swaps: Market Structures, Applications, and Credit Risk," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 61-95.

5. "CFO Forum: Concerns About Counterparties," Institutional Investor (March 1991), p. 144.

6. I. Cooper and A. Mello, "The Default Risk of Swaps," Journal of Finance (June 1991), pp. 597-620.

7. W. Glasgall and B. Javetski, "Swap Fever: Big Money, Big Risks," Business Week (June 1, 1992), pp. 102-106.

8. B. Iben, "Interest Rate Swap Valuation," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 266-279.

9. K. Kapner and J. Marshall, The Swaps Handbook, New York, New York Institute of Finance, 1991.

10. I. Kawaller, "Interest Rate Swaps Versus Eurodollar Strips," Financial Analysts Journal (September/October 1989), pp. 55-61.

11. K. Simons, "Measuring Credit Risk in Interest Rate Swaps," New England Economic Review (November/December 1989), pp. 29-38.

12. C. Smith, C. Smithson, and L. Wakeman, "The Market for Interest Rate Swaps," Financial Management (Winter 1988), pp. 34-44.

13. S. Sundaresan, "Valuation of Swaps," Working Paper, Columbia University, September 1989.

14. L. Wall and J. Pringle, "Alternative Explanations of Interest Rate Swaps: A Theoretical and Empirical Analysis," Financial Management (Summer 1989), pp. 59-73.

Appendix A. The Internal Rate of Return on a Mark-to-Market Swap- Linked Debt Structure

In Section II, we noted that because of the trade-off that is created between making (receiving) a swap unwind payment and then adjusting the remainder of the agreement to a lower (higher) fixed swap rate, the internal rate of return is virtually invariant to the exact swap rate that prevails on each future settlement date. Unfortunately, the multiperiod nature of the cash flow discounting involved in the mark-to-market process renders a generalized closed-form proof of this statement impossible. The essence of these relationships, however, can be captured analytically. We first present a two-period simplification of the example summarized in Exhibit 2.

Specifically, suppose a company borrows NP dollars for two periods at an interest rate [I.sub.t-1] that is reset at dates 0 and 1 and paid in arrears. If this company first considers entering into a "plain vanilla" swap as the payer of the fixed rate F in exchange for [I.sub.t-1], then its net periodic swap cash flows on dates 1 and 2 are (F - [I.sub.0])(NP) and (F -[I.sub.1])(NP), respectively. Consequently, the internal rate of return (i.e., funding cost) on this borrowing can be established by solving:

NP = [[I.sub.0] + (F - [I.sub.0])](NP)/[(1 + IR[R.sub.p]).sup.1] + [[I.sub.1] + (F - [I.sub.1])](NP) + (NP)/[(1 + IR[R.sub.p]).sup.2]

= F(NP)/(1 + IR[R.sub.p]) + (1 + F)(NP)/[(1 + IR[R.sub.p]).sup.2] (A1)

for IR[R.sub.p]. In this case, IR[R.sub.p] = F as the swapped borrowing is equivalent to a synthetic fixed-coupon bond issued at par value.

Alternatively, if this company enters into a mark-to-market swap agreement, on date 1 the original contract will be unwound and replaced with a new one at the rate prevailing for the last period. Letting (F + X) represent this new swap fixed rate, which is also assumed to be the discount rate for the unwind payment, the total swap-related cash flows are now given by [(F- [I.sub.1])(NP) - #(F + X) - F#(NP)(1 + F + X).sup.-1] and [(F + X - [I.sub.2])(NP)], respectively. Notice that the first of these cash flows (i.e., for date 1) includes both a regular settlement payment on the original swap and an unwind payment, while the second (i.e., for date 2) is just to settle the replacement swap. With these amounts, the cost of funds using the mark-to-market structure (i.e., IR[R.sub.m]) can be found by solving:

NP = [F - (X)[(1 + F + X).sup.-1]](NP)/[(1 + IR[R.sub.m]).sup.1] + (1 + F + X)(NP)/[(1 + IR[R.sub.m]).sup.2]. (A2)

Letting [Theta] = (1 + F + X) and recognizing that NP can be dropped from both sides of the equation, Equation (A2) can be rearranged as:

[(1 +IR[R.sub.m]).sup.2] - [F- (X)[([Theta]).sup.-1]](1 +IR[R.sub.m]) - ([Theta]) = 0

which, of course, is a straightforward polynomial in (1 + IR[R.sub.m]). As such, it can be solved for IR[R.sub.m] as follows:

[Mathematical Expression Omitted]

Notice that IR[R.sub.m] [is not equal to] F; that is, the internal rate of return on the mark-to-market structure is not strictly identical to the initial swap rate. From Equation (A3), however, it can be established that the difference between IR[R.sub.m] and F is quite small. For instance, using the positive root, if F = 8% and X = 0.5%, IR[R.sub.m] = 8.001%; for F = 6% and X = 1%, [IRR.sub.m] = 6.005%. (Incidentally, the source for this small difference rests with the reinvestment assumptions built into the internal rate of return measure and is why in practice swap unwind payments are usually discounted using the zero-coupon rates discussed in footnote 9.)

Given that the two-period solution in Equation (A3) cannot be solved definitively for IR[R.sub.m], generalizing this algebraic analysis further is not fruitful. For example, from Equation (4) we have seen that the date T settlement payment with the mark-to-market contract is equal to:

[Mathematical Expression Omitted]

which, when calculated for 0 [is less than] T [is less than or equal to] N and placed into an equation such as Equation (A2), creates a complex polynomial relationship. What we can indicate, though, is how the internal rate of return calculated for the four-period example in Exhibit 2 would change under various alternative -- and quite extreme -- interest rate environments. The chart below shows the value of IR[R.sub.m] that would obtain under several different combinations of: (i) an initial swap rate, and (ii) subsequent movements in replacement swap rates:

[TABULAR DATA OMITTED]

Finally, notice that this display calculates IR[R.sub.m] under the assumption that replacement swap rates always move in one direction. In the present context, this represents a "worst case" scenario; replacement swap rates that vacillated around the initial level would tend to keep IR[R.sub.m] even closer to the initial level of F.

The explosive growth in the swap market over the past decade has masked one vital concern about these agreements. Simply put, though it is generally regarded as a tool for reducing interest rate risk, an interest rate swap is itself a risky transaction. Aggrawal [2] summarized the various types of exposure borne by the swap user, including risk due to unexpected changes in market prices and regulatory, legal, and accounting matters. But the most important of these risks is the potential economic loss that a firm would incur if its counterparty defaulted on the swap agreement when interest rates have moved adversely. Financial managers have now become aware of the default costs associated with using interest rate swaps, which, according to Cooper and Mello [6], are likely to be subordinate to debt claims in the event of bankruptcy. As an indication of this, the same Institutional Investor survey reported that roughly 80% of the CFOs responding expressed increasing concern over the creditworthiness of their swap partners, with over one-third of that group admitting to having actually rejected a potential counterparty because of this risk.

It is difficult to get accurate data on default losses in the swap market because many transactions apparently are worked out without technical default. Kapner and Marshall [9] reported results from a 1987-1988 ISDA survey indicating that only 11 of 71 swap dealers had experienced write-offs, amounting to just $33 million on aggregated portfolios at that time of $283 billion. While this figure suggests very low default rates, it pre- dates the court-ordered default on swap positions held by British local authorities (i.e., municipalities) whereby a number of bank counterparties stand to lose amounts estimated at (pounds)500 million on positions that reached a maximum notional principal of over [pounds]10 billion. In that case, the courts ruled that the local authorities had exceeded their legal power because the swap positions could not be justified as hedges and thus were considered outright speculation. Further, although a 1992 ISDA survey indicated that only about 0.5% of the outstanding notional principal over the last decade was lost to default, Glasgall and Javetski [7] noted that workouts were also needed to transfer positions involving the failed institutions, Bank of New England and Drexel Burnham Lambert, with the latter's swap book totaling over $30 billion.

There are two ways of viewing swap default risk: (i) actual exposure, which is a measure of the loss if the counterparty were to default and is based on the movement in swap market rates between the inception of the agreement and the current date; and (ii) potential exposure, which is based on a forecast of how market conditions might change between the present and the swap's maturity date, including in some manner the probability of default by the counterparty. Thus far, the academic literature investigating this topic has concentrated on estimating the value of potential exposure and showing how it is shared between the counterparties. Simons [11] demonstrated how many of the ad hoc estimation techniques used in practice could be improved by employing simulations of different future interest rate environments. This approach was generalized by Sundaresan [13] who assumed a stochastic process to characterize future interest rate movements in order to show how default risk impacts swap valuation. Cooper and Mello [6], in addition to producing a theoretical model for valuing potential default risk, examined the transfer of this risk between three groups associated with a firm: shareholders, debtholders, and the swap counterparty. While this focus on potential exposure is important, what remains largely unaddressed is the extent of the actual default risk that accumulates as a swap position seasons. The goal of the present research is to explore ways in which the structure of the swap agreement can be revised to reduce this type of exposure.

The analysis is organized as follows. In the first section, we develop an analytic framework to measure actual default exposure and discuss the market practices widely used to deal with that risk. Section II outlines the mechanics of the mark-to-market swap, the first of two design innovations we examine. We demonstrate that the mark-to-market swap agreement is essentially a sequence of swap commitments, each of which is liquidated after a single settlement period. A second design innovation, the forward rate swap, is introduced in Section III. In this approach, the usual swap structure is modified at the time of its origination by altering the pattern of fixed swap rates to reflect expectations about the market conditions that will prevail on the future settlement dates. Examples of both designs, along with a discussion of practical issues in their implementation, are presented in each respective section. In the final section, we offer some concluding comments.

I. Default Risk on an Interest Rate Swap

A. Measuring Default Risk Exposure

Default risk exposure on an interest rate swap differs from that on an ordinary bond in three important ways. First, the most common (i.e., "plain vanilla") swap structure requires no exchange of principal payments at inception or maturity, establishing instead a notional principal used to convert interest rates into cash flows for each settlement period. Therefore, there is no principal at risk. Second, a swap is an executory contract, signifying that one party need perform only if the counterparty also performs. If one party ceases paying the fixed rate, the other need not pay the floating rate. Combined, these two considerations indicate why the risk exposure on a swap is significantly less than that on a bond of comparable face value. In fact, the exposure is limited to the difference between the present values of the remaining fixed-rate and floating-rate cash flows. Because that net present value may be positive or negative, the third important difference between a swap and a bond is that the default risk exposure is bilateral in that each party must be concerned about the possible default of the other.

To measure default risk, suppose that on date-T (i.e., T periods ago, relative to the current date 0) Firms A and B agreed to a swap transaction wherein Firm A makes payments based on a prespecified fixed rate every settlement date in exchange for receipt of cash flows based on a variable interest rate index. Settlement typically is on a net basis, the difference between the fixed and floating flows. This transaction, assumed to mature at date N, will then have a total of T + N settlement dates over its lifetime. Let [F.sub.-T, N + T] be the fixed rate in the transaction (denoting the rate established at date -T for a swap with T + N periods to maturity) and [I.sub.t-1] be the index rate observed at date t - 1 and payable at date t. Swaps typically settle in arrears in that the floating rate is set at the beginning of the period and payment is made at the end.

Now suppose that immediately after the settlement payment (or receipt) on date 0, Firm B enters bankruptcy proceedings, thereby defaulting on the remaining N settlement obligations. Assume that on date 0 the current market fixed rate on a swap with N periods to maturity is [F.sub.0, N], a rate assumed to be higher than [F.sub.-T, N + T]. As the payer of what had been a "below- market" fixed rate, Firm A experiences a loss equal to the following amount:

[Mathematical Expression Omitted]

where [V.sub.0] is the economic value of the swap on date 0, [r.sub.t] is the (zero-coupon) risk-adjusted discount rate appropriate for each of the remaining N cash flows, and NP is the notional principal on the swap. The value of the swap to Firm A on date 0 is the discounted value of the difference between the cash flows generated by the higher fixed rate on a replacement swap ([F.sub.0, N]) and its original level ([F.sub.-T, N + T]). If [V.sub.0] were negative, implying that [F.sub.0, N] is less than [F.sub.-T, N + T], Equation (1) would represent the amount that Firm A would gain if its counterparty were to default. In that case, Firm B would be receiving an above- market fixed rate for payment of the floating-rate index; the swap would represent a valuable asset rather than a liability to be extinguished in bankruptcy. Clearly, Firm B would choose not to default on the agreement, rather it would prefer to continue the transaction or attempt to transfer it to a third party at a profit.

Firm A's actual risk exposure (RE) following default by Firm B at date 0 can then be summarized by:

[Mathematical Expression Omitted]

Similarly, as the fixed-rate receiver, Firm B's risk exposure at date 0 is:

[Mathematical Expression Omitted]

Equations (2) and (3) are the mark-to-market risk exposures for Firms A and B in that they summarize the loss that would be incurred based on the current swap market rate. Notice that the exposures depend on the joint occurrence of two events: (i) the default by the counterparty, and (ii) an adverse movement in interest rates. While potential default risk at origination may be bilateral, because of this second condition the actual exposures at any subsequent date will be unilateral. That is, it will never be the case that both parties simultaneously have positive mark-to-market exposures to one another.(1)

How default risk exposure evolves over time depends fundamentally on the replacement swap fixed rate, [F.sub.0, N], and the time remaining in the contract, N. Generally, the more that [F.sub.0, N] diverges from the existing rate, [F.sub.-T, N+T], and the larger is N, the greater is the default loss. Early in the life of the swap, when N is high but [F.sub.0, N] is likely to be relatively close to [F.sub.-T, N+T], the exposure would be relatively small. At an extreme, if the counterparty enters bankruptcy proceedings immediately after signing the swap document, the default loss is likely to be negligible given that the firm presumably can replace the swap at the same market rate. Late in the life of the swap when N is low, the exposure again can be small despite a significant spread between [F.sub.0, N] and [F.sub.-T, N+T]. In fact, at maturity, the risk exposure cannot exceed the difference between the floating-rate index at the last settlement date ([I.sub.N-1]) and the fixed rate, times the notional principal. Overall, mark-to-market risk exposure is likely to reach an internal maximum at some time between the origination and maturity dates.

B. Market Practices to Deal With Default Risk

Most swap market makers are either commercial or investment banks. Because of the fundamental difference in the nature of their business, however, they deal with default risk on swaps differently. Commercial banks are in the credit-risk-bearing business -- that is their chief function and source of market expertise.(2) As such, they have tended to price default risk explicitly into the fixed rate on the swap, raising their receive-fixed (offer) rate and lowering their pay-fixed (bid) rate as compensation to the extent that the counterparty represents additional risk. As would be expected, the presence of greater risk widens the market maker's bid/offer spread. Similarly, entering a swap with a stronger counterparty would induce the bank to narrow its bid/offer spread. Investment banks, on the other hand, are not in the business of bearing the credit risk of customers over the long-term. Consequently, they have tended to use letters of credit and posting of collateral to raise the effective credit standing of weaker counterparties to their level. Rather than pricing the default risk of a particular firm into the fixed rate on the swap, investment banks would have a uniform bid/offer spread applicable only to acceptable counterparties.

The key point is that it is the imbalance in degrees of credit risk between the counterparties that motivates the adjustment in the fixed rate on the swap or the request for collateral (or letter of credit). Given that each counterparty must assess its default risk exposure to the other, neither would be very willing to give up something in the agreement without getting the same in return. So, collateral is used in the swap market mostly to equalize a disparity in the creditworthiness of the counterparties, not to manage the inevitable exposure that arises even between two comparable firms.(3) Mutual exchange of collateral presents the problem of retrieving the securities that had been posted if the counterparty were to default on the swap, especially if those securities had been lent out in the "repo" market. An alternative would be to post collateral with a third party acting as a clearinghouse. The value of securities to be posted could be a fraction of the notional principal or the market value of swap. In any case, the use of collateral raises transactions costs in the same manner as the margin account on an exchange-traded futures contract.

Because of the bilateral nature of potential credit risk, swaps are negotiated on a more "level playing field" than typically encountered with bank loan contracts. That, in turn, also limits the ability of both parties to demand restrictive covenants. However, there are two such items that usually are contained in swap documentation. The first is a cross-default clause which triggers the swap's default in the event that any indebtedness of one of the counterparties is either in default or has been accelerated. This provision mitigates the effects of what we describe below as a default-timing option. Second, swap agreements also often require that multiple contracts between the same counterparties be netted against each other, which eliminates the ability of the defaulting party to "cherry-pick" contracts -- i.e., default on only those with negative economic value while maintaining the ones with positive value. To guarantee that provisions such as these are adopted, market makers prefer that customers sign a master swap agreement to expedite future transactions and ensure that all existing swap positions can be netted out if a default were to occur.

A default-timing option can arise if the defaulting party can benefit by postponing the event of default itself. Suppose that the economic value of the swap, as measured by Equation (1), is negative but the firm is scheduled to receive a payment at the next settlement date. For example, assume Firm B is in serious financial distress at a time when [F.sub.0, N] [is greater than] [F.sub.-T, N+T] and [I.sub.0] [is less than] [F.sub.-T, N+T]. This means that the existing receive-fixed rate is below-market even though the next scheduled settlement provides a net cash inflow. This could occur if the yield curve dramatically steepened after the swap was originated, driving the short-term index rate down and the longer-term fixed rate up. Firm B clearly would have no reason to default on the swap prior to the next receipt. In general, it would have an incentive to defer defaulting on the swap as long as the floating-rate index calls for net cash inflows. Moreover, all creditors of Firm B would concur with the decision to defer default until a time when [F.sub.0, N] [is greater than] [F.sub.-T, N+T] and [I.sub.0] [is greater than] [F.sub.-T, N+T].(4) As noted, a cross-default clause works against such a default-timing option by identifying as events of default factors other than direct nonpayment of an owed amount on a settlement date; for instance, filing of bankruptcy proceedings by any other creditor will also bring the swap into default, triggering termination of the agreement.

II. Swap Design Innovations: Mark-to-Market Swaps

A. Revising the Structure of the Swap Agreement

We have shown that any firm engaged in an interest rate swap will incur a loss if its counterparty defaults at a time when interest rates have moved in an adverse direction. One way to eliminate this exposure is to have the participants liquidate their existing agreement every time interest rates change and then renegotiate the terms of a new swap to reflect the prevailing market conditions. Of course, because interest rates are never static this solution is impractical in that it would require virtually perpetual exchanges of mark-to-market cash flows. As a more manageable alternative, the counterparties could agree to a scheme in which the renegotiations occur only on the periodic settlement dates of the swap (i.e., when cash flow exchanges were scheduled to occur anyway). That is, imagine that Firms A and B had structured the following agreement at date-T:

(i) Enter into a plain vanilla swap for T + N periods with Firm A paying, and Firm B receiving, the fixed rate of [F.sub.-T, N+T];

(ii) On the first settlement date (i.e., date -T + 1), the counterparties: (a) make their respective fixed- and floating-rate payments on the existing swap; and (b) liquidate the remaining portion of the existing swap according to Equation (1) above, using the current swap rate as the discount factor for all future cash flows, i.e., set [r.sub.t] equal to [F.sub.-T+1, N+T-1], which would be the fixed rate as of date -T + 1 on swap having N + T - 1 periods until maturity;

(iii) Immediately upon unwinding the old swap, the participants enter into a new swap having the same notional principal as the original but (N + T - 1) periods to maturity. The new fixed rate on this swap, again, would be [F.sub.-T+1, N+T-1];

(iv) Repeat steps (ii) and (iii) on each settlement date until the maturity of the original swap (i.e., date N).

This procedure, though admittedly contrived, captures the essence of the mark-to-market swap transaction. Formally, rather than having the counterparties enter and then unwind (N + T - 1) different swaps -- as is implied by the above scheme -- the mark-to-market swap would be a single agreement made at date -T and terminating at date N. The difference between it and the plain vanilla commitment covering the same period is that while the latter generates default risk exposure between dates -T and 0 by leaving the fixed rate set at [F.sub.-T, N+T], the mark-to-market swap eliminates this exposure on each settlement period. It does this by having the counterparties settle in cash any change in the swap's economic value and then resetting the fixed rate on the remainder of the agreement to the prevailing level. Thus, Firm A's settlement payment at date 0 can be written:

[Mathematical Expression Omitted]

where, as noted above, the new swap fixed rate is used as the common discount factor for all future cash flows. Notice that Equation (4) consists of two parts, the first being the scheduled net settlement payment on the existing swap and the second being the present value of unwinding the remainder of that swap according to Equation (1).

Beyond its ability to reduce default risk, another key property of this "marking" procedure is that the overall internal rate of return on a swap-linked variable rate funding structure is relatively invariant to the exact swap rate that prevails on each future settlement date. This follows directly from the trade-off that is created between making (receiving) a swap unwind payment and then adjusting the remainder of the agreement to a lower (higher) rate. Consequently, no matter what path interest rates follow in the future, the swap unwind payment -- and subsequent resetting of the fixed rate on the "new" swap -- always link the process back to the conditions in the market at date 0. This point is illustrated by the following example and documented more thoroughly in the Appendix.

B. A Numerical Example

Suppose that a corporation has just issued a two-year, floating-rate note (FRN) with a par value of $10 million, calling for semiannual interest payments based on the London Interbank Offer Rate (LIBOR). Because this firm presumably would prefer a known cost of funds but now would be exposed to rising rates, it can convert its new debt into the equivalent of a fixed-rate issue via the swap market. This can be accomplished by entering a pay-fixed, two-year (i.e., four settlement period) swap against six-month LIBOR with a notional principal of $10 million. That is, as floating-rate payments are required on its FRN, the firm will receive an equal amount from its swap counterparty in exchange for cash flows based on the fixed swap rate. Letting the current fixed rate for such an agreement be [F.sub.0, 4] = 9.00%, Exhibit 2 summarizes the data necessary to illustrate the mechanics of the procedure outlined above.(5)

Panel A specifies one possible pattern for swap fixed rates on each future settlement date. Note that with each successive settlement date, the maturity of the swap is reduced by six months to match the remaining maturity on the underlying note. Panel B shows the calculations of the periodic cash flows associated with these assumptions, including both the fixed payments on the existing swap and the liquidation value of the remainder of that swap.(6) For example, on date 1 the economic value of the existing agreement is [V.sub.0] = -$69,049, a negative amount since the firm is obligated to pay 9.00% when the current market rate is only [F.sub.1, 3] = 8.50%. The corporation would have to pay its counterparty that amount to close out the existing agreement. In addition, the fixed payment due on that date is $450,000, given the initial fixed rate of 9.00%. Finally, Panel C computes the corporation's synthetic funding cost, expressed as the internal rate of return of the combined FRN/swap transaction.

This example highlights two important attributes of the mark-to-market swap. First, and most importantly, notice that when measured to the basis point the internal rate of return on the aggregate position is 9.00%. Of course, this is the same net funding rate that the firm would have had if it could have found a counterparty willing to accept 9.00% as a fixed rate on a plain vanilla swap with otherwise comparable terms. Thus, when viewed solely in terms of the cost of funds over the entire two years, the plain vanilla and mark-to-market swap structures achieve the same result. Second, unlike the plain vanilla swap, which would have required constant settlement payments of $450,000, the mark-to-market scheme generates highly variable cash flows from period to period. The exact payment pattern will depend on the path of the fixed rates on the sequence of replacement swaps, which in practice could come from the market maker's own quote sheet or as the median of a set of quotes from competing swap dealers. We should stress, though, that the plain vanilla swap keeps the periodic cash flows constant at the expense of creating default risk exposure for one counterparty or the other. The whole point of the mark-to-market structure is to eliminate this exposure on each settlement date by converting it to an obligatory cash payment.

C. Practical Considerations

Mark-to-market swaps have been traded in some form since 1987. Originally created by Manufacturers Hanover Trust Company as a means of giving less creditworthy firms access to the swap market, the product has gained slow, but steady, acceptance. Two difficulties often are cited by practitioners as barriers to implementing this innovation. First, many counterparties lack either the expertise or the back office technical support necessary to track the cash flow changes during the life of the agreement. In such cases, the only alternative is to rely exclusively on the calculations of the swap dealer, something many firms are understandably reluctant to do. This is a particular concern for a sizable portion of the professional (i.e., bank) counterparty market that has found it difficult to adapt existing accounting and computing systems to handle a large volume of anything other than plain vanilla transactions.

A second reason for the reluctance of many corporate counterparties to use the mark-to-market scheme stems from the unpredictability of the cash flow streams it creates. More precisely, this swap structure might obligate the counterparty to make sizable unwind payments prior to maturity. For instance, at date 1 in the above example, the firm had to pay almost $70,000 more than it expected based on information available at date 0. A company that actually has to raise this money in the capital market may find little consolation in the fact that its settlement payments will then be lowered for the remainder of the swap. This is a particular concern to industrial corporations who may not access their capital sources with the same frequency as do financial firms. Further, if the firm does have to tap external funding sources, its internal rate of return will only equal the original fixed rate on the plain vanilla agreement if its borrowing cost is equal to the prevailing fixed swap rate used as a discount factor. However, if the firm's cost of capital exceeds [F.sub.0, N], its aggregate funding cost from the bond/swap combination will exceed [F.sub.-T, N+T].(7)

On the other hand, participants in the swap market have found two features of the mark-to-market structure to be quite attractive. The most appealing attribute, of course, is the ability to mitigate the actual default risk exposure to their swap counterparties. In principle, it is possible to reduce this exposure even further by marking the agreement more frequently, perhaps using a system of margin accounts to collateralize the adjustment process. Practically, though, this variation creates measurable costs for the firm in that additional resources would have to be committed to monitoring its position. Further, it is likely to violate the guidelines created by the Commodity Futures Trading Commission to keep the swap and exchange-traded futures markets distinct. The second desirable feature of the mark-to-market format is that neither counterparty is hurt by a miscalculation of the new fixed swap rate on each settlement date. That is, if the reset rate agreed to on date t is ([F.sub.t, N-t] + [Epsilon]), as opposed to its "true" level of [F.sub.t, N-t], the fixed-rate payer would then be required to make higher future settlement payments but receive the present value of this amount as an unwind payment at date t. Again, this stresses that the pattern of future reset rates -- whether driven by market forces or human error -- is not by itself an important factor.

III. Swap Design Innovations: Forward Rate Swaps

A. Departures from the Plain Vanilla Swap Structure

The inherent structure of a plain vanilla interest rate swap, whereby a single fixed rate applies to all settlement dates, is itself a source of default risk. This is because in virtually all yield curve environments, there will be actual default risk exposure associated with a given contract even if interest rate movements are nonstochastic. In other words, if the future path for the floating-rate index were fully deterministic, a firm would typically incur a financial loss if its swap counterparty defaulted. The exceptional circumstance when this might not occur is an extended period of constant interest rates and a totally flat yield curve. That rarity aside, default risk arises because interest rates are stochastic, a reality exacerbated by the setting of a uniform fixed rate for the lifetime of the swap.

Consider again the fixed rate ([F.sub.0, N]) on a new N-period swap set on date 0. For both counterparties to the swap, the sequence of expected future payments and receipts is determined by the sign of [F.sub.0, N] - E([I.sub.t-1]). However, assuming risk-neutral participants, the swap market will be in equilibrium only if the present value of the fixed flows set by [F.sub.0, N] equal the present value of the expected variable flows set according to [I.sub.t]. This, in turn, means that the swap has an initial economic value of zero to each counterparty if the following condition holds:

[Mathematical Expression Omitted]

Note from Equation (5) that the initial exchange at the end of the first period is based on the known, current level of the floating- rate index ([I.sub.0]) while subsequent exchanges are only known to the level of their expected values.

Suppose that the overall market environment is one of expected rising short-term rates, reflected by an upward-sloping yield curve. This means that as the single swap fixed rate, [F.sub.0, N], will be a complex average of the yields expected to prevail during the N-period life of the agreement. Consequently, for the fixed- payer (fixed-receiver) on the swap, there is likely to be some future date t* after (before) which the firm expects to be receiving cash settlement payments and before (after) which it expects to be making payments. To the fixed-payer, the default risk exposure is then "back-loaded" in that it would be more concerned about its counterparty defaulting in the later years of the agreement when receipts are anticipated. For the fixed-receiver, on the other hand, the default risk would be front-loaded because it expects to be receiving settlement payments early and to be paying later. These conclusions, of course, implicitly rely on the expectations theory of the yield curve and the parties to the swap having subjective expectations aligned with the overall market.

A departure from the structure of a plain vanilla swap to address this front-loading and back-loading problem would be to set a time-varying fixed rate on the swap. That is, the fixed rate would be prespecified for all future settlement periods at inception but it would differ for each period so as to minimize expected settlement payments and receipts, thereby minimizing the expected default risk. For example, suppose that the uniform fixed rate is replaced in Equation (5) with the set of implied forward rates known at date O, based on the swap yield curve for maturities ranging from one to N periods.

[Mathematical Expression Omitted]

where IF[R.sub.0, 1] is the observed rate (e.g., LIBOR) for a transaction between dates 0 and 1 and IF[R.sub.t-1, t] is the implied forward rate on a one-period security between dates t -1 and t. An agreement structured in this fashion is what we call a forward rate swap. It is not a forward swap in the sense of a having a deferred starting date, rather it contains a sequence of forward fixed rates for future exchanges. This sequence need not be limited to just implied forward rates calculated from spot market quotations; it could come from observed rates in the explicit forward or futures market. For example, Eurodollar futures and over-the-counter forward rate agreement (FRA) markets could be used to establish the sequence.(8) In fact, the set of rates could simply be mutually acceptable to the two counter parties based on their own subjective probabilities of future rates.

The reduction in default risk exposure depends critically on the extent to which the actual path for the floating-rate index follows the trajectory of the forward rates. If the difference between the two is small, default risk can be substantially reduced. For instance, if IF[R.sub.t-1, t] is a reasonably good measure of E([I.sub.t-1]), the expected payoff for each future settlement date should be less than setting a single fixed rate. Note that the implied forward rate need not be a perfect predictor of future market rates for default risk exposure to be reduced as long as the variance of (IF[R.sub.t-1, t] - [I.sub.t-1]) is less than the variance of ([F.sub.0, N] - [I.sub.t-1]). However, it should be emphasized that if interest rates move significantly away from the implied forward path, the default risk can be even higher than on a plain vanilla swap structure. So, unlike the mark-to-market swap, a forward swap might not always reduce default risk.

B. A Numerical Example

Suppose that the fixed rates on one-, two-, and three-year annual settlement interest rate swaps (against twelve-month LIBOR) are [F.sub.0, 1] = 8%, [F.sub.0, 2] = 10%, and [F.sub.0, 3] = 11%, respectively. Current twelve-month LIBOR is also eight percent, so the one-year swap quote really only provides a benchmark -- it would be an agreement to both pay and receive the same known rate. Thus, if a firm enters the three-year swap as the fixed-rate payer at the market rate of 11%, it knows in advance that it is scheduled to make a settlement payment in one year equal to [F.sub.0, 3] - [I.sub.0] = 3% times the notional principal. Similarly, the fixed-rate receiver is assured of the receipt of the same amount, barring default by the counterparty. Clearly, the market levels for LIBOR in one and two years (i.e., [I.sub.1] and [I.sub.2]) will determine which counterparty ultimately receives the greater cash inflows. The fixed-rate payer gains if future LIBOR rises sufficiently above 11% in future periods to compensate for the first year's known cash outflow.

The implied forward rates based on this swap yield curve are IF[R.sub.0, 1] = 8% (= [I.sub.0]) for the first year, IF[R.sub.1, 2] = 12.245% for the second, and IF[R.sub.2, 3] = 13.408% for the third.(9) The forward rate swap structure would set the fixed rate for the first-year settlement at 8%, the second-year settlement at 12.245 %, and the third-year at 13.408%. That this swap has the same initial economic value as the plain vanilla structure using a single fixed rate of 11% for each period can be demonstrated by calculating the present values of each fixed cash flow stream:

8%/1.08 + 12.245%/[(1.10102).sup.2] + 13.408%/[(1.11193).sup.3] = 27.261%

= 11%/1.08 + 11%/[(1.10102).sup.2] + 11%/[(1.11193).sup.3]. (7)

While the plain vanilla and forward rate structures have the same initial economic value, the degrees of default risk exposure and the settlement cash flows will differ by a considerable amount. Suppose that a firm agrees to pay the fixed rate on a swap and that LIBOR does follow the implied forward rate trajectory (i.e., twelve-month LIBOR rises from [I.sub.0] = 8% to IF[R.sub.1, 2] = 12.245% on the next settlement date). With a plain vanilla swap, the firm would be obligated on that date to make a settlement payment of three percent times the notional principal because LIBOR at inception was eight percent. In addition to that payment, the swap would have positive mark-to-market value given that the two-year replacement swap fixed rate would be above 11%. With 12-month LIBOR equal to 12.245% on that date and an expected rate for the following year of 13.408%, the replacement swap fixed rate [F.sub.1, 2] can be calculated as follows:

12.245%/1.12245 + 13.408%/[(1.13408).sup.2] = [F.sub.1, 2]/1.12245 + [F.sub.1, 2]/[(1.13408).sup.2]. (8)

Solving for [F.sub.1, 2] obtains 12.726%. The economic value of the swap, and hence the default risk exposure, is the present value of the difference between 12.726% and 11% for the next two settlement dates, an amount equal to 2.875% (times the notional principal). That amount, by the way, is what would be received by the firm in a mark-to-market swap in exchange for resetting the fixed rate to 12.726% for the remaining settlement dates. In sharp contrast, a forward rate swap in the same environment would entail no settlement payment or receipt on that date. Moreover, the default risk exposure would be zero. It follows that this structure minimizes default risk to the extent that the actual trajectory for the floating-rate index follows the path set for the time-varying fixed rate on the swap. What matters is that actual rate movements are positively correlated with the scheduled changes in the fixed rate on the swap.

C. Practical Considerations

While there is an emerging market for mark-to-market swaps, we know of no actual swap transactions that employ the forward rate structure to mitigate default risk exposure. One possible reason for this could be similar back office technical difficulties as with mark-to-market swaps. A time-varying fixed rate would require some monitoring and investment in accounting and information systems, which, while demanding an increased time commitment from the company's treasury unit, would likely only burden the first-time user. The more plausible reason, however, is that swaps between commercial banks and their corporate customers have tended to have the bank receiving, and the corporation paying, the fixed rate. In addition, yield curves have been upward-sloping in recent years, especially in the intermediate-term maturities that typify swap agreements. These two circumstances combined have moderated concern over default risk from the perspective of the bank, but not necessarily so for the corporate end-user. On plain vanilla swap contracts, the corporations that are paying the fixed rates would be making net settlement payments in the early years of the transaction and receiving them in the later years. This is preferable for the bank, since it is much easier to assess the probability of financial distress in the near term than in the more distant future.

When the yield curve is upward-sloping, the proposed forward rate structure could even exacerbate the default risk exposure from the perspective of the commercial banks. In the later years of the swap, when the fixed rate would be higher than if a single rate applied to all periods, a default by a counterparty scheduled to be paying the fixed rate would be more costly to the bank. This suggests the three related conditions for the introduction of forward rate swaps: (i) corporations becoming more leery of banks as counterparties and seeking out swap designs that minimize default risk from their own perspective; (ii) an extended period of inverted yield curves, such as prevailed in the early 1980s in the United States and more recently in the United Kingdom; and (iii) more corporate demand to receive the fixed rate on interest rate swaps.

IV. Concluding Comments

The two design innovations we discuss, mark-to-market swaps and forward rate swaps, both draw the structure of the agreement closer to that of an exchange-traded futures contract. Futures exchanges manage their default risk by daily mark-to-market valuation and settlement transactions posted to a margin account. That this is done on a daily basis owes to the key role played by "locals" -- individual traders who provide much of the liquidity to the market but are thinly capitalized compared to institutional participants, commercial and investment banks, and major corporations. The fact that plain vanilla swaps do not typically require margin accounts with daily settlement (or, more generally, any form of collateralization) has been central in their appeal to corporate managers of financial risk. So these innovations, which would introduce periodic mark-to-market settlement and time-varying fixed rates, borrow from the futures market, yet still maintain a separate identity.

Default risk arises because firms do fall into financial distress and market interest rates are stochastic. The mark-to-market design for the swap uses actual rates to make ex post adjustments to the fixed rate. The forward rate design uses expected rates to make ex ante adjustments. Each design can effectively transform a floating-rate liability to a fixed-rate liability with a known, locked-in cost of funds subject, of course, to some assumptions about reinvestment rates. However, the amount of the net interest payment for each period is not known in advance with the mark-to-market design. With the forward rate swap, the future amount is known but is likely to differ for each period. The appeal of the plain vanilla structure is that it can provide a constant, known net interest payment for all periods. The trade-off is one of certainty with respect to future cash flows versus default risk. The mark-to-market swap manages default risk most effectively, limiting exposure to rate changes only over the most recent settlement period. The forward rate swap manages the risk less precisely and only to the extent that future rates follow the path structure in the agreement. Market rate changes accumulate on both the plain vanilla and forward rate structures, and can lead to a much larger amount of default risk exposure, given the direction and extent of the rate movements and the time remaining until maturity. Ultimately, application of these innovative designs for swap contracts will depend on economic conditions -- as risks arise, techniques and product designs always emerge to manage those risks.

1 In addition to mark-to-market exposure, which represents the actual consequence of a swap default at a specific point in time, many swap market participants are also concerned with estimating the maximum potential risk over the agreement's lifetime. These counterparties, typically money-center banks concerned about their total credit exposure to a corporate client, often determine a measure called fractional exposure. Although we have found that procedures for calculating fractional exposure vary across institutions, it is essentially an estimate of the worst-case scenario for mark-to-market risk between the present date and the swap's maturity. Thus, fractional exposure is analogous to the potential default risk measure analyzed by Simons [11] and Cooper and Mello [6].

2 A motivation for commercial banks to enter and promote the swap market has been to generate off-balance-sheet income and to exploit their comparative advantage in credit-based products at a time when traditional lending to corporate customers was falling off. Brown and Smith [3] described this as the "reintermediation" of commercial banking.

3 This follows, in part, from the idea advanced in Brown and Smith [4] that there will be a common fixed rate for all swap transactions involving counterparties of comparable creditworthiness. That is, in equilibrium, two AA-rated firms will negotiate to the same fixed rate as two BBB-rated credits.

4 There also is a plausible scenario whereby Firm B would elect not to default even though both [F.sub.0, N] [is greater than] [F.sub.- T, N+T] and [I.sub.0] [is greater than] [F.sub.-T, N+T]. Depending on the level of [I.sub.0], the firm could choose to make a small net settlement payment to keep the swap alive, in hope that swap market fixed rates subsequently fall and the swap takes on positive economic value. In effect, the periodic settlement payment is the premium for an out-of-the-money option on an interest rate swap. The value of that option might exceed its purchase price.

5 It should be noted that this example is typical of how corporations participate in the swap market. In fact, Wall and Pringle [14] provided survey data indicating that large companies are five times more likely to be fixed-rate payers than fixed-rate receivers. This same study also contains an excellent discussion of the many motivations for using swaps.

6 In these calculations, two simplifying adjustments have been made to Equation (4). First, the scheduled settlement payment is listed on a gross basis, rather than net of the LIBOR-based swap receipt. This was done because the floating-rate swap receipt will merely be passed through to the firm's bondholders and can therefore be ignored. Second, all interest rates were altered to reflect semiannual payment periods even though they were originally expressed on an annual basis.

7 A similar statement can be made for the unwind funds received on each settlement date. If these receipts cannot be reinvested at [F.sub.0, N], then the actual internal rate of return for the combined transaction will not be equal to [F.sub.-T, N+T].

8 This is consistent with the observation, expressed in Smith, Smithson, and Wakeman [12] and Kawaller [10] among other places, that plain vanilla interest rate swaps can be replicated to some degree by strips of either over-the-counter forward or exchange-traded futures contracts on the index interest rate.

9 To calculate the implied forward rates, one first needs to derive (or observe in the market) the zero-coupon rates that are consistent with the prevailing yield curve on coupon securities. In this context, the phrase "consistent with" means that there would be no arbitrage profits available from buying the coupon bond and selling its coupon and principal cash flows as separate zero-coupon instruments, nor from buying zeros and "reconstituting" the coupon bond. Iben [8] has provided a discussion of this technique and its application in the swap market. For example, the two-year zero-coupon rate of 10.102% is found as the solution for [Z.sub.2] in: 100 = 10/1.08 + 110/[(1 + [Z.sub.2]).sup.2]. The par value purchase price of the two-year bond must equal the present value of the future cash flows. Similarly, the three-year rate of 11.193% is found as the solution for [Z.sub.3] in: 100 = 11/1.08 + 11/[(1.10102).sup.2] + 111/[(1 + [Z.sub.3]).sup.3]. Notice that the first-year's coupon flow of 11 is discounted by the known eight percent one-year rate and the second year's flow is discounted by 10.102%, the result of the previous calculation. The implied forward rates are then calculated in the usual manner as: IF[R.sub.1, 2] = [[(1.10102).sup.2]/(1.08)] - 1 = 0.12245 and IF[R.sub.2, 3] = [[(1.11193).sup.3]/[(1.10102).sup.2]] - 1 = 0.13408.

References

1. P. Abken, "Beyond Plain Vanilla: A Taxonomy of Swaps," Federal Reserve Bank of Atlanta Economic Review (March/April 1991), pp. 12-29.

2. R. Aggrawal, "Assessing Default Risk in Interest Rate Swaps," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 430-448.

3. K. Brown and D. Smith, "Recent Innovations in Interest Rate Risk Management and the Reintermediation of Commercial Banking," Financial Management (Winter 1988), pp. 45-58.

4. K. Brown and D. Smith, "Plain Vanilla Swaps: Market Structures, Applications, and Credit Risk," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 61-95.

5. "CFO Forum: Concerns About Counterparties," Institutional Investor (March 1991), p. 144.

6. I. Cooper and A. Mello, "The Default Risk of Swaps," Journal of Finance (June 1991), pp. 597-620.

7. W. Glasgall and B. Javetski, "Swap Fever: Big Money, Big Risks," Business Week (June 1, 1992), pp. 102-106.

8. B. Iben, "Interest Rate Swap Valuation," in Interest Rate Swaps, C. Beidleman (ed.), Homewood, IL, Business One-Irwin, 1990, pp. 266-279.

9. K. Kapner and J. Marshall, The Swaps Handbook, New York, New York Institute of Finance, 1991.

10. I. Kawaller, "Interest Rate Swaps Versus Eurodollar Strips," Financial Analysts Journal (September/October 1989), pp. 55-61.

11. K. Simons, "Measuring Credit Risk in Interest Rate Swaps," New England Economic Review (November/December 1989), pp. 29-38.

12. C. Smith, C. Smithson, and L. Wakeman, "The Market for Interest Rate Swaps," Financial Management (Winter 1988), pp. 34-44.

13. S. Sundaresan, "Valuation of Swaps," Working Paper, Columbia University, September 1989.

14. L. Wall and J. Pringle, "Alternative Explanations of Interest Rate Swaps: A Theoretical and Empirical Analysis," Financial Management (Summer 1989), pp. 59-73.

Appendix A. The Internal Rate of Return on a Mark-to-Market Swap- Linked Debt Structure

In Section II, we noted that because of the trade-off that is created between making (receiving) a swap unwind payment and then adjusting the remainder of the agreement to a lower (higher) fixed swap rate, the internal rate of return is virtually invariant to the exact swap rate that prevails on each future settlement date. Unfortunately, the multiperiod nature of the cash flow discounting involved in the mark-to-market process renders a generalized closed-form proof of this statement impossible. The essence of these relationships, however, can be captured analytically. We first present a two-period simplification of the example summarized in Exhibit 2.

Specifically, suppose a company borrows NP dollars for two periods at an interest rate [I.sub.t-1] that is reset at dates 0 and 1 and paid in arrears. If this company first considers entering into a "plain vanilla" swap as the payer of the fixed rate F in exchange for [I.sub.t-1], then its net periodic swap cash flows on dates 1 and 2 are (F - [I.sub.0])(NP) and (F -[I.sub.1])(NP), respectively. Consequently, the internal rate of return (i.e., funding cost) on this borrowing can be established by solving:

NP = [[I.sub.0] + (F - [I.sub.0])](NP)/[(1 + IR[R.sub.p]).sup.1] + [[I.sub.1] + (F - [I.sub.1])](NP) + (NP)/[(1 + IR[R.sub.p]).sup.2]

= F(NP)/(1 + IR[R.sub.p]) + (1 + F)(NP)/[(1 + IR[R.sub.p]).sup.2] (A1)

for IR[R.sub.p]. In this case, IR[R.sub.p] = F as the swapped borrowing is equivalent to a synthetic fixed-coupon bond issued at par value.

Alternatively, if this company enters into a mark-to-market swap agreement, on date 1 the original contract will be unwound and replaced with a new one at the rate prevailing for the last period. Letting (F + X) represent this new swap fixed rate, which is also assumed to be the discount rate for the unwind payment, the total swap-related cash flows are now given by [(F- [I.sub.1])(NP) - #(F + X) - F#(NP)(1 + F + X).sup.-1] and [(F + X - [I.sub.2])(NP)], respectively. Notice that the first of these cash flows (i.e., for date 1) includes both a regular settlement payment on the original swap and an unwind payment, while the second (i.e., for date 2) is just to settle the replacement swap. With these amounts, the cost of funds using the mark-to-market structure (i.e., IR[R.sub.m]) can be found by solving:

NP = [F - (X)[(1 + F + X).sup.-1]](NP)/[(1 + IR[R.sub.m]).sup.1] + (1 + F + X)(NP)/[(1 + IR[R.sub.m]).sup.2]. (A2)

Letting [Theta] = (1 + F + X) and recognizing that NP can be dropped from both sides of the equation, Equation (A2) can be rearranged as:

[(1 +IR[R.sub.m]).sup.2] - [F- (X)[([Theta]).sup.-1]](1 +IR[R.sub.m]) - ([Theta]) = 0

which, of course, is a straightforward polynomial in (1 + IR[R.sub.m]). As such, it can be solved for IR[R.sub.m] as follows:

[Mathematical Expression Omitted]

Notice that IR[R.sub.m] [is not equal to] F; that is, the internal rate of return on the mark-to-market structure is not strictly identical to the initial swap rate. From Equation (A3), however, it can be established that the difference between IR[R.sub.m] and F is quite small. For instance, using the positive root, if F = 8% and X = 0.5%, IR[R.sub.m] = 8.001%; for F = 6% and X = 1%, [IRR.sub.m] = 6.005%. (Incidentally, the source for this small difference rests with the reinvestment assumptions built into the internal rate of return measure and is why in practice swap unwind payments are usually discounted using the zero-coupon rates discussed in footnote 9.)

Given that the two-period solution in Equation (A3) cannot be solved definitively for IR[R.sub.m], generalizing this algebraic analysis further is not fruitful. For example, from Equation (4) we have seen that the date T settlement payment with the mark-to-market contract is equal to:

[Mathematical Expression Omitted]

which, when calculated for 0 [is less than] T [is less than or equal to] N and placed into an equation such as Equation (A2), creates a complex polynomial relationship. What we can indicate, though, is how the internal rate of return calculated for the four-period example in Exhibit 2 would change under various alternative -- and quite extreme -- interest rate environments. The chart below shows the value of IR[R.sub.m] that would obtain under several different combinations of: (i) an initial swap rate, and (ii) subsequent movements in replacement swap rates:

[TABULAR DATA OMITTED]

Finally, notice that this display calculates IR[R.sub.m] under the assumption that replacement swap rates always move in one direction. In the present context, this represents a "worst case" scenario; replacement swap rates that vacillated around the initial level would tend to keep IR[R.sub.m] even closer to the initial level of F.

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Title Annotation: | Security Design Special Issue; includes appendix |
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Author: | Brown, Keith C.; Smith, Donald J. |

Publication: | Financial Management |

Date: | Jun 22, 1993 |

Words: | 8881 |

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