CHAPTER 14 SWAPS. To examine the reasons for undertaking plain vanilla, interest rate and currency swaps.

Transcription

1 1 LEARNING OBJECTIVES CHAPTER 14 SWAPS To examine the reasons for undertaking plain vanilla, interest rate and currency swaps. To demonstrate the principle of comparative advantage as the source of the mutual gains in a swap to all parties. To examine the role of the swap dealer, settlement procedures and pricing schedules. To show how interest rate and currency swaps can be valued by creating a replication portfolio consisting of either a position in bonds or a in series of forward contracts. To examine the key features of more complex swap agreements (eg. basis, diff and rollercoaster swaps). Swaps are privately arranged contracts (ie. OTC instruments) in which parties agree to exchange cash flows in the future according to a pre-arranged formula. Swap contracts originated in about The largest market is in interest rate swaps but currency swaps are also actively traded. The most common type of interest rate swap is a plain-vanilla or fixed-for-floating rate swap. Here one party agrees to make a series of fixed interest payments to the counterparty, and to receive a series of payments based on a variable (floating) interest rate. The payments are based on a stated notional principal, but only the interest payments are exchanged. The payment dates and the floating rate to be used (usually LIBOR) are also determined at the outset of the contract. In a plain vanilla swap the fixed rate payer knows exactly what the interest rate payments will be on every payment date but the floating rate payer does not. It may be immediately obvious to some readers that an interest rate swap is (analytically) nothing more than a series of forward rate agreements, FRA s (see Cuthbertson and Nitzsche 2001). As we shall see, one method of pricing the swap is to use implied forward-forward rates to calculate the

2 2 value today of the uncertain future variable rate cash flows. Since the swap is equivalent to a series of FRA s (or forward contracts) then what swaps offer is lower transactions costs than the series of FRA s. Also if there is an element of oligopoly in lending institutions so that credit spreads on direct borrowing are relatively high then swaps provide a method of circumventing this problem, providing a net gain to all parties in the swap (assuming no one defaults). The intermediaries in a swap transaction are usually banks who act as swap dealers. They are usually members of the International Swaps and Derivatives Association (ISDA) which provides some standardization in swap agreements via its master swap agreement and this can then be adapted where necessary, to accommodate most customer requirements. Dealers make profits via the bid-ask spread and might also charge a small brokerage fee. If swap dealers take on one side of a swap but cannot find a counterparty then they have an open position (ie. either net payments or receipts at a fixed or floating rate). They usually hedge this position in futures (and sometimes options) markets until they find a suitable counterparty INTEREST RATE SWAPS A swap can be used to alter series of floating rate payments (or receipts) into fixed rate payments (or receipts). Consider a corporate that has issued a floating rate bond and has to pay LIBOR+0.5 (figure 14.1(A)). If it enters a swap to receive LIBOR and pay 6% fixed, then its net payments are 6% + 0.5% = 6.5% fixed. It has transformed a floating rate liability into a fixed rate liability. Figure 14.1(B) shows how the issue of a 6.2% fixed rate bond plus a receive 6% fixed, pay LIBOR floating results in a floating rate liability at LIBOR + 0.2%. Similarly, if the corporate (or a financial institution) holds a 5.7% fixed rate bond (figure 14.2(A)) and enters a pay 6% fixed, receive LIBOR floating swap then the net result is an asset which pays floating at LIBOR 0.3%. Finally, if the financial institution holds a floating rate bond

3 3 paying LIBOR-0.5% (figure 14.2(B)) and enters a receive LIBOR floating, pay 6% fixed then the net result is a receipt of 5.5% fixed. Now let us see how a swap can be used to reduce interest rate risk of a financial institution. The normal commercial operation of some firms naturally imply that they are subject to interest rate risk. A commercial bank or Savings and Loan in the US (Building Society in the UK) usually has fixed rate receipts in the form of loans or housing mortgages, at say 12% but raises much of its finance in the form of short-term floating rate deposits, at say LIBOR - 1% (figure 14.3). If LIBOR currently equals 11% the bank earns a profit on the spread of 2% pa. However, if LIBOR rises by more than 2% the S&L will be making a loss. The financial institution is therefore subject to interest rate risk. If it enters into a swap to receive LIBOR and pay 11% fixed, then it is protected from rises in the general level of interest rates since it now effectively has fixed rate receipts of 2% which are independent of what happens to floating rates in the future. A second reason for undertaking a swap is that some firms can borrow relatively cheaply in either the fixed or floating rate market. Suppose firm-a finds it relatively cheap to borrow at a fixed rate but would prefer to ultimately borrow at a floating rate (so as to match its floating rate

4 4 receipts). Firm-A does not go directly and borrow at a floating rate because it is relatively expensive. Instead it borrows (cheaply) at a fixed rate and enters into a swap where it pays floating and receives fixed. This cost saving is known as the comparative advantage motive for a swap. We consider this case below. Suppose it is the 15th of March and two firms face the following situation. Firm-A is currently borrowing $100m at a fixed rate but would prefer to borrow floating. Firm-B is currently borrowing $100m floating but would prefer to borrow at a fixed rate. To achieve their preferred borrowing pattern, the two firms can agree to swap interest payments. In the swap (figure 14.4) one party agrees to receive known fixed interest payments at predetermined dates in the future and to pay out at a set of floating rates, which are unknown at time t=0 when the swap is initiated (except for the 1 st payment, see below). Suppose the reference rate for the floating payment is 6-month LIBOR and on the 15 th of March, firm-a enters into a pay floating, receive fixed swap over 1 year. Firm-A has then agreed to pay firm-b LIBOR on the 15th of September and the 15th of March (of the next year) and to receive fixed rate payments from firm-b. Hence, in the swap: PLAIN VANILLA SWAP : Firm-A is a floating rate payer and a fixed rate receiver and therefore Firm-B is a floating rate receiver and a fixed rate payer.

5 5 Since firm-a originally borrowed funds at a fixed rate but now receives fixed rate payments in the swap and pays floating rate payments in the swap, then firm-a is effectively ends up paying a floating rate. Similarly, since firm-b originally borrowed funds at a floating rate but now receives floating rate payments in the swap and pays at a fixed rate in the swap, it is effectively ends up paying at a fixed rate. Floating rate payments are determined by LIBOR at the beginning of each (6-month) reference period (figure 14.5). Hence the known value of LIBOR on 15th March of say 11% is used to determine the floating payment due on the 15th September, and whatever LIBOR turns out to be on the 15th of September will determine the floating payment due on the 15th March (of the next year). At the same time, namely 15th March, firm-b agrees to pay firm-a a fixed rate (r x ) of say 10% on each of the two reference dates. Note that no exchange of principal takes place only the difference in interest payments are exchanged. Day count conventions differ but we assume (for simplicity) that all 6-month periods comprise 180 days and one-year has 360 days. In the swap, firm-b is a floating rate receiver at LIBOR and a fixed rate payer at 10%. The cash flow received by firm-b at each payment date is : Firm-B is Floating Rate Receiver and Fixed Rate Payer Receipts of Firm-B = $100 [LIBOR 0.10](180/360) Notice that the swap receipts are determined in a very similar way to those for an FRA. Clearly firm-b will have net receipts if LIBOR rates exceed the fixed rate. At the time the swap contract is first agreed on the 15th of March, the net cash flow to be paid on the 15th September is known (since the known LIBOR rate on the 15th of March is the reference rate). However,

6 6 neither party knows on the 15th March what the net payments will be in 1-years time, since this depends on the actual value of LIBOR on the 15th of September. One possible outcome is depicted in figure On the 15th of March firm-b knows it will receive the difference between the known floating rate of 11% and the fixed rate of 10% on $100m notional principal, hence firm-b receives $5,000 on the 15th of September (from firm-a). The outturn value for LIBOR on the 15th of September we assume is 10% which happens to equal the fixed rate. Hence the next March, no payments change hands. Overall firm-b has a net receipt of $5,000 (and firm-a had a net payment of $5,000). Of course had LIBOR, on the 15th September, been above 10%, then firm-a would have paid cash to B. PLAIN VANILLA SWAP Interest rate swaps are undertaken because there are net reductions in the cost of borrowing for both parties to the swap. The swap dealer can also appropriate some of these gains. Again, suppose : Firm-A wishes to end up borrowing $10m at a floating rate for 5 years and Firm-B wishes to end up borrowing at a fixed rate for 5-years. Table 14.1 : Borrowing Rates Facing A and B Fixed Floating Firm-A (A x ) LIBOR + 0.3% (A F ) Firm-B (B x ) LIBOR + 1.0% (B F ) Absolute difference (B-A) (Fixed) = 1.2 (Float) = 0.7 Net Comparative Advantage or Quality Spread Differential NCA = (Fixed) - (Float) = 0.5 B has comparative advantage in borrowing at a floating rate. Hence Firm-B borrows at a floating rate. The rates offered to firm-a and firm-b are shown in table A has an absolute advantage in both markets since it can borrow floating and fixed at a lower rate than firm-b (possibly because firm-b has an overall lower credit rating in both markets). Nevertheless there is a net gain to both parties if they enter a swap agreement as can be seen from the following: (1) Total cost to firm-a and firm-b of direct borrowing in preferred form

7 7 = B X + A F = 11.2% + (L + 0.3%) = L % (2) Total Cost to A+B if they borrow in non-preferred form = A X + B F = 10% + (L + 1%) = L + 11% Hence the total cost is lower if they initially borrow in their non-preferred form: Net overall gain to firm-a and firm-b = (B X + A F ) - (A X + B F ) = 0.5% Although there is a reduction in total cost using strategy (2) there is currently a big problem namely, it results in firm-a and firm-b not having their preferred form of borrowing. However, the swap provides the mechanism to achieve the latter and lower the cost of borrowing for both parties. Looking at the overall gain in a slightly different way (table 14.1), the key element is that firm-b has ''comparative advantage'' in the floating rate market, while firm-a has comparative advantage in the fixed rate market. (Comparative advantage is used in international trade theory to help explain why the UK exports wine to France (and vice versa), even though the latter has an absolute cost advantage in producing wine at low cost.) Firm-B has comparative advantage in the floating rate market because firm-b pays only 0.7% more in the floating market than does firm-a, whereas firm-b pays (a larger) 1.2% more than firm-a in the fixed rate market. (If you like, firm-b pays less more in the floating market than in the fixed rate market). Hence firm-b initially borrows floating and firm-a borrows fixed. They then enter into a swap agreement whereby firm-b agrees to pay firm-a at a fixed rate and firm-a pays firm-b at a floating rate, so they both ultimately achieve their desired type of borrowing (ie. firm-b pays fixed and firm-a floating). The net comparative advantage or quality spread differential is : NET COMPARATIVE ADVANTAGE / QUALITY SPREAD DIFFERENTIAL NCA = Difference in Fixed Rate - Difference in Floating Rate = (B X - A X ) - (B F - A F ) = (11.2% - 10%) - (LIBOR + 1%) - (LIBOR + 0.3%)] = 1.2% 0.7% = 0.5% which is the total gain from the swap, noted earlier. CASE 1 : FIRM-A AND FIRM-B DEAL DIRECTLY WITH EACH OTHER We will arbitrarily assume that the gain of 0.5% is split equally (0.25%) between firm-a and firm-b. (This split will depend on the relative bargaining power of firm-a and firm-b). Firm-B

8 8 has a comparative advantage in the floating rate market and hence issues $10m floating rate debt at LIBOR + 1%. Firm-A issues fixed rate debt at 10% (figure 14.6). Here s how we work out the figures in the swap. We initially consider the swap from firm- B s point of view (who ultimately wants to borrow floating) and expects to gain 0.25% overall. (We will find that this ensures that firm-a also achieves again of 0.25%.) This is what happens to firm- B: (1) Firm-B initially borrows direct at floating LIBOR + 1% (in which it has NCA) (2) Assume firm-b in leg1 of the swap agrees to receive LIBOR (3) Net payment by firm-b so far are 1% (fixed) (4) But firm-b must end up with a gain of 0.25%, and hence firm-b s total fixed interest payments must be: we = direct cost of borrowing fixed - swap gain = 11.2% % = 10.95% (5) Hence in leg 2 of the swap firm-b must pay 10.95% - 1% = 9.95% Cash Flows of firm-b : Initially issues floating rate debt and in the swap receives floating and pays fixed pays LIBOR + 1% receives LIBOR from firm-a pays 9.95% fixed to firm-a Hence firm-b ends up paying 10.95% fixed (= 9.95% + 1%) even though it started out by issuing floating rate debt. Although firm-b pays 10.95% fixed, this is 0.25% less than if it went directly to the fixed rate market (at 11.2% fixed - see table 14.1). Does the above allow firm-a to also gain 0.25 from the swap? The cash flows for firm-a are :

9 9 Cash Flows of firm-a : Initially, issues fixed rate debt and in the swap receives fixed and pays floating pays 10% fixed receives 9.95% fixed from firm-b pays LIBOR to firm-b Hence firm-a ends up paying LIBOR % (= LIBOR + 10% %). Firm-A has converted or swapped its fixed interest bond issue into a purely floating rate payment. Also firm- A s floating rate payment is 0.25% less than it would pay if it went directly to the floating rate market where it has to pay LIBOR + 0.3% (see table 14.1). Hence in the swap, firm-a agrees to pay firm-b at LIBOR and firm-b agrees to pay firm-a fixed at 9.95% (figure 14.6). The overall payments and receipts are : Firm-B issues floating at LIBOR + 1% Firm-A issues fixed at 10% Firm-A agrees to pay firm-b at 6m LIBOR on a notional $10m Firm-B agrees to pay firm-a at 9.95% p.a. fixed on notional $10m From the above we can see that the gain from comparative advantage of 0.5% is split evenly between firm-a and firm-b. (Of course this need not necessarily always be the case.) Both firm-a and firm-b gain by 0.25% each, compared with borrowing directly in their preferred form of debt (ie. either fixed or floating). CASE 2 : SWAP DEALER ACTS AS FINANCIAL INTERMEDIARY Assume that the swap dealer takes part of the total gain due to comparative advantage, of 0.5%. In figure 14.7 we assume the swap dealer breaks even on the floating rate, since she pays out and receives LIBOR. On the fixed rate the swap dealer receives 10% but only pays out 9.9%, which provides an overall gain of 0.1% for the swap dealer.

10 10 Cash Flows of firm-a Firm-A initially issues fixed rate debt and in the swap receives fixed and pays floating pays 10% fixed (as above) receives 9.9% fixed from the bank (ie. less than 9.95% - above) pays LIBOR to the bank (as above) The net effect is that firm-a pays LIBOR + 0.1% (which is still 0.2% less than going directly to the floating rate market). From figure 14.6 it is easy to work out firm-b s position. Cash Flows of firm-b Firm-B initially issues floating rate debt and in the swap receives floating and pays fixed pays LIBOR + 1% (as above) receives LIBOR from the bank (as above) pays 10% fixed to the bank (ie. greater than the 9.95% above) The net effect is that firm-b pays 11% fixed (which is 0.2% better than going directly to the fixed rate market). The swap dealer gains 0.1% on the difference between its receipts and payments on the fixed rate deal. Note that the swap dealer is subject to potential default risk since either firm-a or firm-b could default, yet the bank has to honour its commitment to the other party. Also note that LIBOR in the above example can take on any value and the swap deal will still be worthwhile. SWAP DEALER A swap dealer will also usually take on one-side of a swap even if she cannot immediately find a counterparty. This is known a "warehousing". If the dealer does warehouse a swap then she is exposed to interest rate risk which she will usually hedge with a series of interest rate futures contracts. Usually Eurodollar futures will be used and a series of Eurodollar futures is know as a Eurodollar strip. Consider for example figure If the swap dealer initially takes on a swap only with firm-a then she receives LIBOR and pays 9.9% fixed. If LIBOR falls below 9.9% before the swap dealer finds firm-b then she will make a loss. (This is sometimes also referred to a mismatch risk.) It can hedge any single swap payment with firm-a by going long (ie. buying) short-term interest rate futures with a maturity close to that specific payment date. If interest rates fall, the futures price will rise and the profit from the futures offsets the loss on the floating rate leg of the swap.

11 11 In practice the swap dealer will aggregate her overall swap positions with all her clients and decide for each payment date whether she is net long or short on the floating leg of the swaps. For each payment date she can then set in place a Eurodollar futures hedge. In our above stylised example, once the swap dealer finds the counterparty, firm-b, then she is perfectly hedged with certain net receipts of 0.1% (ie. the mismatch risk is eliminated). Note however, that it may be difficult for the swap dealer to find a counterparty-b who has exactly the reverse wishes of firm-a. For example, if the maturity of firm-a s swap is say 5 years but firm-b will only enter into the swap for 3 years then the swap dealer is a net floating rate receiver in years 4 and 5 and is subject to interest rate risk in the last two years of the swap. She may then initially hedge in the (rather illiquid) 4 and 5 year (Eurodollar) futures contracts and probably continue to search for another counterparty for these payments. SETTLEMENT AND PRICE QUOTES Settlement procedures are similar to those for FRA's. Only interest payments are exchanged (not the principal sums). Suppose the reference period is 6-months and the notional principal in the swap (Q) is $10m. LIBOR rates will be known at the beginning of each 6-month leg of the contract but payments are not made until the end of this 6-month period. Suppose 6- month LIBOR is L t-1 = 11% at the end of the first 6-months. Then payments at the end of 12- months, given the rates quoted above, will be (see figure 14.7) A pays to bank: $Q (L t-1 r X,A) 0.5 = ( ) ($10m) 0.5 = $55,000 Bank pays to B: $Q (L t-1 r X,B ) 0.5 = ( ) ($10m) 0.5 = $50,000 Let us take the case of a US swap trader who wishes to set rates for a 5 year swap with the floating rate based on 6-month LIBOR (table 14.2). She will have an indicative pricing schedule (for that day) where the fixed rate will be based on the current 5-year Treasury note (bond) plus the spread. (In fact this will usually be the par bond yield.) If the swap trader agrees to pay fixed and receive floating then she will quote 8.46% (= fixed rate + 46 bp swap spread) and receive 6-month LIBOR. The swap spread reflects the normal credit risk as perceived by the dealer. Notice that no floating rates appear in the pricing schedule of table 14.2 and when this occurs the floating rate is usually understood to be 6-month LIBOR flat. One practical point to note is that LIBOR is quoted assuming semi-annual payments with a 360 day year while US T- notes use semi-annual payments but with a 365 day year. Hence, the LIBOR rate must be multiplied by (365/360) to put it on an equivalent basis to the T-bond rate : T-bond equivalent rate (for a LIBOR quoted rate) = LIBOR x (365/360)

12 12 Table 14.2 : Indicative Pricing Schedule for Swaps Maturity Current T-Bond rate Bank pays fixed Bank receives fixed 4-years yr. T-B + 40 bp 4 yr. T-B + 50 bp 5-years yr. T-B + 46 bp 5 yr. T-B + 56 bp 6-years yr. T-B + 58 bp 6 yr. T-B + 68 bp The dealer will eventually hope to match the fixed rate deal and receive 8.56% fixed (= 8% + 56 bp) thus obtaining a 10 bp bid-ask spread as profit. The bid-ask spread reflects several factors namely, the degree of competition between dealers, the risk in (temporarily) holding an open position on one leg of the deal, and their current inventory position of being either net long or short in the fixed or floating legs of all its outstanding swaps (ie. the overall position of its swap book). The bid-ask rates on interest rate swaps in various currencies are given in figure All the fixed rates are quoted against the appropriate LIBOR rate.(usually 3-month or 6-month). As you can see the bid-ask spreads are rather small, reflecting the high degree of competition and liquidity in the market. Hedging the market and credit risk in the swap book is a key issue for the swap dealer. As far as market risk is concerned she will continually monitor the sensitivity of her swap book to small or large changes in interest rates including any offseting hedges that are in place using futures and options. Hedging is mainly done with Eurodollar futures. Hedging with options is possible but this requires frequent rebalancing and the choice of whether to hedge some or all of the sources of risk (eg. delta, gamma and vega hedges). As we see in later chapters, the concept of Value at Risk is now widely used as a measure of the market risk of a portfolio of assets,

13 13 including swap, futures and options. It attempts to measure the maximum loss on the portfolio that will occur over a specific horizon (eg. one day) with a given probability (say 95% of the time). Note that in a swap between a dealer and company-a only one party will face default risk at any one time (although that party might change over time). This is because at time t>0 if interest rates have changed and the value of the swap to the dealer is V D > 0 then V A must be less than zero. It is only if company-a defaults that there is a problem for the dealer. If the dealer defaults then company-a is not harmed since it can simply reneague on its payments (although in practice it may often not reneague). Credit risk in a fixed for floating swap (or other OTC transactions in forwards and options) difficult to assess but it is often managed by insisting on so called credit enhancements which seek to offset some of the overall default risk. The most common method to limit credit risk is to pledge some sort of collateral such as a line of credit from a bank or a batch of securities which are held in trust. As the market value of the swap alters then so can the amount of collateral. Similarly, if one party undergoes a downgrade in its Standard and Poor or Moody s rating, the collateral can be increased. A variant on this is marking-to-market, whereby from time to time the swaps value is ascertained and one party pays the other this amount in cash. The fixed rate on the swap is then reset to give a zero current value for the swap or the swap may be terminated. Clearly this procedure is similar to margin payments in a futures contract, albeit without the use of a clearing house. Netting is a fairly simple form of credit enhancement. If at t>0 the dealer s swap position with company-a is plus $5m and on another contract is minus $4m then they can agree that the outstanding net position is $1m. Hence if company-a defaults then the swap dealer is only exposed to $1m credit risk rather than $5m. Although it must be pointed out that in the event of bankruptcy by company-a it is not always clear that the bankruptcy courts will legally certify such a deal. It is worth noting that the UK House of Lords deemed that UK Local Authorities (ie. equivalent to US Municipal authorities) acted ultra vires (ie. beyond the scope of authority) in entering into swap contracts and these contracts then became null and void (even though the Local Authorities had the funds to close out their swaps losses). This legal ruling in the UK accounts for about ½ of the credit losses on swaps, todate. Total losses from swap defaults have historically been very low (eg less than about ½% of the principal value of all swaps entered into). This is probably because swap deals tend to involve large well capitalised organisations with a high credit rating and reputation to preserve.

14 14 One of the most recent proposals to measure (as opposed to offset) the changing credit risk of OTC instruments (including swaps) is the so called CreditMetrics approach which uses changes in Standard and Poor s credit ratings to provide a measure of the change in a counterparty s credit risk. We discuss this at length in chapter 25, together with other models of credit risk. Also in chapter 25 we outline the use of credit derivatives as a means of hedging the credit risk of swaps. TERMINATING A SWAP Suppose a swap agreement has been in existence for some time and the current value of the swap to firm-b who receives LIBOR and pays fixed, is $100,000. Firm-B can terminate the swap by sale or assignment. The firm simply finds a third party to take over the fixed payments and LIBOR receipts and firm-b sells the swap for $100,000. The swap dealer (ie. the counterparty to firm-b) would have to approve the third party. An alternative is for firm-b to undertake a reversal, that is enter a new swap where the cash flows exactly offset the cash flows in the original swap. Finally, firm-b could use a buy-back whereby the original counterparty pays firm-b $100,000 and the swap is annulled. Many firms use swaps in a speculative fashion. If they feel interest rates will rise over several periods into the future, they can enter a swap agreement to pay fixed and receive a floating rate. Although futures and FRAs can achieve similar features to swaps, the latter have very low transactions costs VALUATION OF INTEREST RATE SWAPS A swap can be priced by either considering the swap as a synthetic bond portfolio or as a series of forward contracts. Of course, both methods yield the same answer! This is another example of financial engineering whereby a swap is analytically equivalent to another (two) portfolios, either of bonds or of futures contracts. When a swap is initiated it will have a value to each party of zero. However, over time its value to any one party can be positive or negative as the PV of the fixed payments in the swap alter as interest rates change. Somewhat paradoxically the PV of the variable rate payments remain largely unchanged even though the (coupon) payments themselves alter. It is this latter proposition that we need to establish first and since the floating rate payments on a swap are equivalent to a floating rate note (FRN) we should know how to value the latter. For those readers interested in a blow by blow account this is done in the appendix 14.1 but here we present a heuristic argument to obtain the key valuation results on the FRN and then we move straight to valuing the swap.

15 15 VALUING AN INITIAL SWAP CONTRACT A floating rate bond (note) is one whose coupon payments are adjusted in line with prevailing market interest rates, which are determined at the previous coupon date. Consider a notional principle of $Q and a LIBOR rate of L 0 at time t=0. At t=0 the coupon payment payable at t=1, on the floating rate bond is known and equals C 1 = L 0 Q. Subsequent coupon payments on the floater depend on future LIBOR rates L 1, L 2, which are not known at t=0. However, it is shown in appendix 14.1 that even though these future coupon payments are uncertain, nevertheless the following are true: 1) At inception, all the receipts on a floating rate bond have a value equal to the notional principal, or par value Q. 2) Immediately after a coupon payment on a floating rate bond, its value also equals the par value Q. If coupon payments were continuously adjusted to changes in interest rates, the price of the floater would always be equal to its par value. Between coupon payment dates, interest rates generally do not change drastically so that the price of the floater does not deviate (too much) from its par value Q. This simplification enables us to value a swap contract. To price a swap at the outset means finding that value of the fixed rate in the swap which makes the swap have zero initial value. For example, suppose you are a swap dealer who has to decide the fixed rate to charge in a new fixed for floating (LIBOR) swap on a notional principal of $25m. The swap will be for 2 years with payments every 180 days and the term structure of LIBOR is 12% p.a. over 6 months, 12.25% p.a. over 1-year, 12.75% over 18 months and 13.02% p.a. over 2 years (all rates are continuously compounded and we assume a 360 day year). The value of the floating leg at t=0 is equal to its par value of $25m. Hence: To price the swap means to calculate the (fixed) coupon which makes the value of the fixed leg in the swap equal to $25m (and express this as a simple annual coupon rate). We have: 25 = C e C e ( 0.5)0.12 (1) C e (1.5) C e e (2) (2) = C

16 16 Hence, the semi-annual fixed rate coupon which satisfies the above equation is C = $ m giving an annual coupon rate (cp) of 13.39% (= C x 2/Q = x 2/25) per $1 nominal. Hence the swap dealer will make zero expected profit if she sets the fixed leg of the swap at a rate of 13.39%. If the dealer is a fixed rate receiver and floating rate payer she will set the actual rate above 13.39% to reflect the market risks, transactions cost (of hedging the position) and the credit risk of the floating rate payer. PRICING SWAPS USING A SYNTHETIC BOND PORTFOLIO At time t>0 the swap can have either a positive of negative value to one of the parties. The remaining cash flows in an interest rate swap can be replicated by positions in two bonds. In this way we create a synthetic swap and use our ideas on the pricing of bonds to establish the value of a swap. If you are a swap dealer (eg. a bank) who is receiving floating and paying fixed your net cash receipts at each 6-month reference date on $100m nominal principal are $100m (L t- 1 r X ) (180/360). But this is the same cash flow position that would ensue from being short (ie. issuing) $100m in a fixed coupon bond and using the proceeds to purchase (ie. go long) a floating rate bond. At the outset there is no exchange of cash (just as in the case of a swap). At maturity, the floating rate bond is redeemed at par which (notionally) provides the funds to pay the $100m face value of the fixed rate bond. We have created a synthetic swap using fixed and floating rate bonds. Consider a swap dealer (eg. a bank) who pays a floating rate and receives a fixed rate. The cash flows for the swap dealer are depicted in our earlier figure 14.4, where the dashed lines indicate (unknown) floating rate payments and the solid lines the known receipts from the fixed rate payments. The value of the swap to the dealer is the difference between the present value of the fixed receipts and the floating payments. At the outset of the swap deal the value of the swap is zero : ex-ante, each party to the swap expects no net gain (ignoring the bid-ask spread). After some time has elapsed the swap will have positive value to the swap dealer if floating rates have fallen (and vice versa). Let : V = value of the swap BX = value of the fixed rate bond underlying the swap BF = value of the floating rate bond underlying the swap Q = notional principal value in the swap agreement n = number of 6-month reference dates remaining r i = yield (continuously compounded) corresponding to maturity t i. At the outset of the swap (at t=0) then V = 0, but during the life of the swap, the value to the dealer is:

17 17 [14.1] V = BX BF The fixed $-coupon payments are C = (r x /2)Q every 6-months, where r x is the fixed rate agreed in the swap. If we are 3-months into the life of the swap (at t in figure 14.4), the next payments will be in 3, 9, 15, months and the present value of these fixed rate receipts is : [14.2] BX (at t = 3-months) = i n 1 C e r t i i Q e r t n n where r i are the continuously compounded spot rates and t i = 0.25, 0.75, 1.25, (These discount rate should reflect the risk class of the counterparty in the swap). It is clear from [14.2] that if future LIBOR rates r i (i > 1) rise, then the present value of the fixed rate receipts falls and hence the value of this leg of the swap to the dealer falls. Next consider the floating leg of the swap. If we were valuing the floating payments in the swap immediately after a coupon payment then BF = Q. Between payment dates we use the fact that BF will equal Q immediately after the next payment date. At t=3-months, the time until the next payment date is t 1 = 0.25 in our notation because the next floating rate payment is due 3- months from time t (ie. 6-months from t=0). Discounting these payments back to time t=3- months, into the swap we have (see appendix 14.1, equation A14.7) : [14.3] BF (at t= 3-months) = (Q + C*/2) 1 e - t r 1 where r 1 is the continuously compounded spot rate over the period t+3-months to t+6-months and C* = Q(LIBOR 0 /2) is the first floating rate payment which is known at time t (as it is based on LIBOR at the previous reset date, which here is that at the outset of the swap agreement, at t = 0). It is somewhat paradoxical that although the size of all future floating rate swap payments (apart from the next one) are uncertain, nevertheless the value of all these future payments is known at time t, as can be seen from equation [14.3] where all the right hand side terms are known. The reason for this is that all the future floating rate payments are set to par immediately after each of the payment dates. Hence BF, between payment dates (like t), depends only on the present value of the next known single payment (C*/2 at t = 6-months) plus the par value Q. It is clear from equation [14.2] that if future LIBOR rates r i (i > 1) rise then the present value of the fixed rate receipts falls while the value of floating rate payments in equation [14.3]

18 18 remains unchanged. Hence the overall value of the swap to the dealer falls (and the value of the swap to the counterparty rises). Of course, if the swap dealer pays fixed and receives floating then the value of the swap to the swap dealer is : [14.4] V = (BF BX) An example of how to calculate the value of a swap to a bank who pays floating and receives fixed is shown in table 14.3: Table 14.3 : VALUE OF SWAP: PAY FLOAT AND RECEIVE FIXED Bank pays floating at 6-months LIBOR and receives fixed at 8% pa (semi-annual payments) Remaining life of swap = 1.25 years Payment Periods are at t i = 0.25, 0.75 and 1.25 years Notional principal/ par value Q = $100m Fixed (coupon) payments = ($100m) 0.08 / 2 = $4m LIBOR at last fixing date was 10.2% (semi-annual payments) Hence, next (known) floating rate payment = ($100m) (0.102 / 2) = $5.1m Current spot rates with continuous compounding for 3-month, 9-month and 15-month, maturities are r 1 = 10%, r 2 = 10.5% and r 3 = 11.0%. Then Value of Fixed Receipts at t = 1.25 years: n BX = i 1 C e r t i i Q e r t n n = (0.10) 0.75(0.105) 1.25(0.11) e 4e 104e = $ r 1 BF = (Q + C*) 1 t e = ( ) -0.25(0.10) e = $102.51m Value of the swap for receive fixed- pay floating = BX BF = -$4.27m

19 19 SWAPS AS A SERIES OF FORWARD CONTRACTS A swap involves net payments or receipts at various known time periods (see figure 14.4). At the outset of the swap (t=0) these net payments have zero value. However, if the swap has been in existence for some time, then at time t (> 0), it may have a positive or negative value, depending on what has happened to interest rates since the swap was initiated. The net payments in a swap are like the payoffs from a series of forward contracts (often referred to as a strip). If you hold a futures contract to maturity then it is equivalent to a forward contract. For example, if you are long a 3-month Eurodollar futures with a yield of 8% (on a notional principal of $1m) which matures in one-years time. This contract locks in a rate of 8% over 3-months, beginning in one years time. If interest rates fall over the coming year to say 7%, the futures price will rise and at maturity a gain of $2,500 (=100 bp x tick value of $25) will be made. A single fixed payment in a swap is like the initial futures/forward (delivery) price and the uncertain floating rate swap payment corresponds to the unknown closing price of the futures contract. Because a swap comprises a series of future net payments it is like a series of futures/forward contracts, each one maturing on a payment date of the swap. If a swap is equivalent to a strip of forward/futures contracts then why bother having swaps at all? The reasons are because of the lower transactions costs and convenience of swaps. For example, a swap dealer provides confidentiality and anonymity between counterparties, whereas a deal in the futures market is transparent. In contrast to futures contracts, swaps are OTC instruments and can be tailor made in terms of notional principal and timing of payments (although they also involve credit risk and are difficult or expensive to unwind). To create a synthetic swap using futures requires futures contracts with very long maturity dates and often the market for most futures is rather thin (or non-existent) at maturities which exceed two years. An exception here are Eurodollar futures which extend to maturities of over 5-years. The longer maturities for this contract are due to the fact that US swap dealers hedge their outstanding net floating rate swap commitments using this contract and this creates a more active market for long maturity Eurodollar futures. Technically, this is a cross hedge since the floating leg of the swap is priced off LIBOR and the underlying in the futures contract is the Eurodollar rate. However, the two rates tend to move together (for any given maturity). At time t, the first floating rate receipt C* is known and depends on the initial floating rate at t=0, that is L 0. Hence, for semi-annual payments C* = Q (L 0 /2). The first fixed rate payment,

20 20 where r X is the agreed fixed rate, is C = Q (r X /2). If the first payment date is at t 1 then the present value of receive floating and pay fixed is : -r 1 [14.5] (C* C) 1 t r 1 e = Q [(L 0 r X )/2] e - t 1 payments are : The future payoffs (ie. for t > t 1 ) for a receive floating-pay fixed swap with semi-annual [14.6] C* t-1 C = Q (L t-1 r X )/2 But this is the payoff from a forward contract which pays $C* t-1 in exchange for a payment or delivery price of $C. The only difference from a standard forward contract is that LIBOR is the rate set 6-months before the actual payment. However, LIBOR is not known at time t for any of the floating rate payoffs, after the first. But we can use implied forward rates (on LIBOR) calculated from any two (appropriate) spot (LIBOR) rates, known at time t, to calculate the markets best estimate of these forward rates. Then we can replace the unknown L t-1 with the appropriate forward rate f i. If f i is the forward (LIBOR) interest rate (based on semi-annual payments) for the 6-months prior to any actual payment date i (i 2) then the best estimate at time t, of any single future floating rate receipt is $C* t-1 = (f i /2) Q. The value today of this single floating rate receipt less the fixed rate payment is equivalent to the value of a long forward contract with forward price F t = (f i /2) Q and an initial delivery price in the contract of F 0 = $C. When we discussed the value of a forward contract we obtained : - r (T t) [14.7] V f,t = (F t F 0 ) e In terms of our swap contract the equivalent synthetic forward position therefore has a value : i [14.8] V i = [(f i /2) Q C] e r i t The total value of the swap is therefore the value of this series of forward contracts : [14.9] V (receive-float, pay-fixed) = n i 1 i (0.5 f i Q - C) r i e t r 1 + (C* - C) - t 1 e Not surprisingly it can be shown that valuing the swap using either the synthetic bond portfolio or as a series of forward contracts gives identical results.

21 CURRENCY SWAPS A currency swap, in its simplest form involves two parties exchanging debt denominated in different currencies. Currency swaps evolved from back-to-back loans and parallel loans which were used to circumvent exchange controls (particularly before the advent of floating exchange rates when there was often a tax placed on foreign currency transactions). A back-to-back loan might involve a US company and a UK company raising finance in dollars and sterling respectively but then agreeing to swap the principal and interest payments. Parallel loans were made by a domestic firm to a foreign firm s subsidiary. For example, a US firm might raise finance in dollars and pass it to a UK subsidiary located in the US, while the UK parent firm would raise sterling finance and pass it on to the US subsidiary located in the UK. Nowadays, one reason for undertaking a swap might be that a US firm ( Uncle Sam ) with a subsidiary in France wishes to raise finance in French francs (FRF) to finance expansion in France. The FRF receipts from the subsidiary in France will be used to pay off the debt. Similarly a French firm ( Effel ) with a subsidiary in the US might wish to issue dollar denominated debt and eventually pay off the interest and principle with dollar revenues from its subsidiary. This reduces foreign exchange exposure. But it might be relatively expensive for Uncle Sam to raise finance directly from French banks and similarly for Effel from US banks, as neither might be well established in these foreign loan markets. However, if the US firm ( Uncle Sam ) can raise finance (relatively) cheaply in dollars and the French firm ( Effel ) in Francs, they might directly borrow in their home currencies so and then swap the payments of interest and principal, with each other. (Note that unlike interest rate swaps where the principal is notional and is not exchanged either at the beginning or the end of the swap, this is not the case for currency swaps). After the swap Effel effectively ends up with a loan in USD and Uncle Sam with a loan in French Francs. The situation is therefore: Uncle Sam ultimately wants to borrow French Francs but finds it cheap to initially borrow in US Dollars. Effel ultimately wants to borrow in US Dollars but finds it cheap to initially borrow in French Francs. They each borrow in their low cost currency and agree to swap currency payments.

22 22 The swap has enabled them to achieve their desired outcome (at the lowest interest cost). Suppose the US-firm Uncle Sam wishes to borrow FRF500m to expand its hamburger chain in France, and the French firm Effel wishes to borrow $100m to expand its chain of spectacle stores in the US. They each agree to borrow in their own currencies and undertake a swap agreement. At the outset of the swap they exchange the principal amounts (figure 14.9). Suppose, the interest rates facing Uncle Sam and Effel are as given in table The currency swap is worthwhile because the difference between the two interest rates facing Uncle Sam and Effel are not the same (table 14.4). Table 14.4 : Borrowing Rates Dollars Fr. Francs Uncle Sam (US firm) 8.0% 11.5% Effel (French firm) 10.0% 12.0% Absolute difference (Effel-Uncle Sam) 2% 0.5% Net comparative advantage or Quality Spread Differential = 2% - 0.5% = 1.5%. Effel has comparative advantage in borrowing FRF. Note: Ultimately Effel wants to borrow USD and Uncle Sam wants to borrow FRF s. This is the motivation for the swap. Effel is less credit worthy than Uncle Sam and has to pay higher rates in both currencies. However, Effel has comparative advantage borrowing FRF, since Effel only pays 0.5% more than Uncle Sam in the French bond market, whereas Effel pays 2% more than Uncle Sam in the dollar market. (Of course this also implies that Uncle Sam has comparative advantage in the dollar market.) From figure 14.8 we have :

23 23 Effel borrows FRF at 12% Uncle Sam borrows USD at 8% They then swap payments because: Uncle Sam ultimately wants to borrow FRF Effel ultimately wants to borrow dollars Since Effel has comparative advantage in borrowing FRF, a swap will lead to lower costs all round. The net comparative advantage (or quality spread differential) is 2% 0.5% = 1.5% (see table 14.4) which can be shared out between Uncle Sam, Effel and the swap dealer. Let us (arbitrarily) assume: Swap dealer gets 0.4% Uncle Sam gets 0.3% Effel gets 0.8% Uncle Sam s gain of 0.3% then implies that it pays = 11.2% on the FRF leg of the swap (whereas it would have had to pay 11.5% directly). Effel s gain of 0.8% implies its dollar payments in the swap are reduced from a direct cost of 10% to 9.2% on the swap (see table 14.4). One possible outcome is given in figure where (for simplicity) the swap dealer is assumed to pay Uncle Sam 8% in dollars and Effel 12% in FRF, so that the two firms payments and receipts are matched. Hence the key elements in the swap of figure are:

24 24 Uncle Sam borrows dollars at 8% but receives dollars at 8% from the swap dealer, making Uncle Sam s net $ payments zero. Uncle Sam then pays the swap dealer FRF at 11.2% (which is cheaper than borrowing directly at 11.5% - see table 14.4). Hence the gain for Uncle Sam is 0.3% (= 11.5% %). Effel borrows FRF at 12% but also receives this from the swap dealer making its net FRF position zero. Effel pays dollars at 9.2% to the swap dealer (which is cheaper than borrowing directly in dollars at 10% - see table 14.4). Hence the gain for Effel is 0.8% (= 10% - 9.2%). The swap dealers position is : Percent dollar gain = 9.2% - 8.0% = 1.2% Absolute dollar gain on $100m = ($9.2m - $8.0m) = $1.2m Percent FRF loss = 12.0% % = 0.8% Absolute FRF loss on FRF500m = ( ) FRF500m = FRF60m - FRF56m = FRF4m Net Percent Gain = 1.2% - 0.8% = 0.4% Overall the total (percentage) gain to all parties in the swap is : [14.10] Uncle Sam + Effel + Swap Dealer = 0.3% + 0.8% + 0.4% = 1.5% which is equal to the "comparative advantage" in table The net gain for the swap dealer involves two currencies: net receipts of $1.2m and net payments of FRF4m. However, for each year of the swap agreement the dealer can hedge this risk using foreign currency forwards or futures. The dealer is also exposed to credit risk. If we had made Effel receive only FRF at 11% from the dealer but it had to pay 12% to its FRF bondholders then some of the foreign exchange risk of the swap dealer would be transferred to Effel. At the maturity date of the swap Uncle Sam has to provide Effel with FRF500m and Effel has to provide Uncle Sam with $100m so they can each pay off their bondholders (ie. the arrows in figure 14.9 are in the opposite direction). In a currency swap the principal as well as interest payments are exchanged VALUATION OF CURRENCY SWAPS

25 25 The net effect of the swap on Uncle Sam is that it converted its $100m of bonds at 8% into a FRF500m bond issue. The swap implies that Uncle Sam receives French Francs at the outset of the swap but then has to pay periodic French Franc interest payments and the FRF500m principal at the end of the swap. Also, Uncle Sam receives periodic dollar interest payments from the swap dealer (figure 14.10) and will receive $100m from the counterparty (ie. Effel ) at the end of the swap. From Uncle Sam s perspective the swap is equivalent to a synthetic position consisting of : Holding (long) a dollar denominated bond and issuing a FRF denominated bond Receives $ s and pays out FRF. If the Dollar-French Franc exchange rate at the outset of the swap is 0.2($/FRF), then the initial exchange of principal of $100m and FRF500m implies the initial value of the swap is zero. The interest rates in the swap are fixed (by assumption) and hence the only uncertainty over the life of the swap from the point of view of Uncle Sam is the future value of the exchange rate. Uncle Sam has a liability in French Franc and hence a future strengthening of the French Franc against the dollar will involve losses on the swap for Uncle Sam (and a gain for Effel ). Payments/Liability in French Franc for Uncle Sam Hence, appreciation of FRF (depreciation of the USD) implies loss on swap. The effect of a future appreciation of the French Franc on the swap dealers open position depends on its asset/liability position in each currency. The swap dealer (figure 14.10) has net dollar interest receipts (of $1.2m) and net liabilities (payments) in French Franc (of FRF4m). Hence, the swap dealer will loose from a strengthening of the French Franc. CURRENCY SWAPS AS A BOND PORTFOLIO From the point of view of Uncle Sam the value of the swap is the difference between a short position in a French Franc bond at a cost of 11.2% p.a. and a long position in a US bond which pays 8% p.a. The spot rates of interest and the current exchange rate are known. Given the coupon payments on one side of the swap, the coupon payments on the other side of the swap are adjusted so that at the inception of the swap (t=0) the value of the swap is zero (to both parties). Now, suppose the swap has been in existence for some time then the value of the swap in dollars at time t is: [14.11] $V = B D - (S)B F