6.47 6.51 6.490
7 6.65 6.68 6.665
10 6.83 6.87 6.850
where the market maker will pay fixed and receive floating; the offer is the fixed rate in a swapwhere the market maker will receive fixed and pay floating. Table 6.5 shows typical quotes for plainvanilla U.S. dollar swaps. As mentioned earlier, the bid—offer spread is 3 to 4 basis points. Theaverage of the bid and offer fixed rates is known as the swap rare. This is shown in the final columnof Table 6.5.
Consider a new swap where the fixed rate equals the swap rate. We can reasonably assume thatthe value of this swap is zero. (Why else would a market maker choose bid—offer quotes centeredon the swap rate?) From equation (6.1), it follows that
Box = Ba (6.2)
As mentioned in Section 5.1, banks and other financial institutions usually discount cash flows inthe over-the-counter market at LIBOR rates of interest. The floating-rate bond underlying theswap pays LIBOR. As a result, the value of this bond, By, equals the swap principal. It followsfrom equation (6.2) that the value of the fixed-rate bond, By, also equals the swap principal. Aswap rate is therefore a LIBOR par yield. It is the coupon rate on the LIBOR bond that causes itto be worth par.
Determining the LIBOR Zero Curve
In Section 5.12 we showed how Eurodollar futures can be used to determine LIBOR zero rates.Swap rates also play an important role in determining LIBOR zero rates. As we have just secn,they define a series of LIBOR par yieid bonds. The iatter can be used to bootstrap a LIBOR zero ©curve in the same way that Treasury bonds are used to bootstrap the Treasury zero curve (seeSection 5.4).
Example 6.1 Assume that the LIBOR zero curve has already been calculated out to 1.5 years(using spot LIBOR rates and Eurodollar futures) and that we wish to use the swap rates in Table 6.5to extend the curve. The 6-month, I-year, and 1.5-year zero rates are, respectively, 5.5%, 5.75%, and5.9% per annum with continuous compounding. Because the swaps in Table 6.5 involve semiannualcash flows, we first interpolate between the swap rates to obtain swap rates at intervals of 0.5 years.The 2.5-year swap rate is 6.135%, the 3.5-year swap rate is 6.2975%, and so on. Next we use thebootstrap method described in Section 5.4. Because the 2-year swap rate is the 2-year par yield,136 CHAPTER 6
a 2-year bond paying a semiannual coupon of 6.045% per annum must sell for par, so that3.0225e-005*05 4 3.922507 00575*1.0 4. 3,9995e-05%15 4 193.0225 7 = 100
where R is the 2-year zero rate. This can be solved to give R = 5.9636%, Similarly, a 2.5-year bondpaying a semiannual coupon of 6.135% must sell for par, so that
3.0675¢7 905505 + 3.0675¢ 2057510 + 3.0675¢ C:059xES 4+ 3.067 5e 7 0-059636 x2 4 103.0675e727>" = 100
where R is the 2.5-year zero rate. This can be solved to get R = 6.0549%. Continuing in this way, thecomplete term structure is obtained. The 3-, 4-, 5-, 7-, and 10-year zero rates are 6.1475%, 6.2986%,6.4265%, 6.6189%, and 6.8355%, respectively.
In the United States, spot LIBOR rates are typically used to define short-term LIBOR zero rates.Eurodollar futures are then used for maturities up to two years—and sometimes for maturities upto five years. Swap rates are used to calculate the zero curve for longer maturities. A similarprocedure is followed to determine LIBOR zero rates in other countries. For example, Swiss francLIBOR zero rates are determined from spot Swiss franc LIBOR rates, three-month Euroswissfutures, and Swiss franc swap rates.
Nature of the LIBOR Zero Curve
The LIBOR zero curve is sometimes also referred to as the swap zero curve. We know that LIBORzero rates out to one year are interbank borrowing rates. It is tempting to assume that the LIBORzero curve beyond one year represents interbank borrowing rates. This is not completely true.
Suppose that the five-year swap rate is 5% and a bank is able to borrow for six months at six-month LIBOR. It can enter into a swap to exchange six-month LIBOR for 5%. This appears tolock in a borrowing rate of 5% for five years. However, as pointed in our criticism of thecomparative-advantage argument, this rate applies only if the bank is able to continue to borrowat six-month LIBOR. In practice the bank is subject to roll-over risk. It may not be able to rollover its borrowings at six-month LIBOR. If the bank’s six-month borrowing rate increases abovesix-month LIBOR, its five year borrowing rate will prove to be greater than 5%. The LIBOR zerocurve is therefore the zero curve for interbank borrowings in a world where the a bank’s interestrate roll-over risk is zero.
6.4 VALUATION OF INTEREST RATE SWAPS
 
An interest rate swap is worth zero, or close to zero, when it is first initiated. After it has been inexistence for some time, its value may become positive or negative. To calculate the value we canregard the swap either as a long position in one bond combined with a short position in anotherbond or as a portfolio of forward rate agreements. In either case we use LIBOR zero rates fordiscounting.
Valuation in Terms of Bond Prices
We saw in equation (6.1) thatVowap = By — Bey
for a swap where floating is received and fixed is paid. In Section 6.3 we used this equation to showSwaps 137
 
that By, equals the swap’s notional principal at the initiation of the swap. We now use it to valuethe swap some time after its initiation. To see how this equation is used, we define:
t;: Time until ith (| <i <n) payments are exchangedL: Notional principal in swap agreementr,: LIBOR zero rate corresponding to maturity t;
k: Fixed payment made on each payment date
The fixed-rate bond, Bs,, can be valued as described in Section 5.3. The cash flows from the bondare k at time t; (1 <i <n) and L at time ¢,, so that
aBix = ke Hi 4 Lene
i=l
Consider next the floating-rate bond. Immediately after a payment date, this is identical to a newlyissued floating-rate bond. It follows that Ba = L immediately after a payment date. Betweenpayment dates, we can use the fact that By will equal L immediately after the next payment dateand argue as follows. Immediately before the next payment date, we have Bg = L +k", where k” isthe floating-rate payment (already known) that will be made on the next payment date. In ournotation, the time until the next payment date is t,;. The value of the swap today is its value justbefore the next payment date discounted at rate r, for time f;:
Bu=(L+k)e7"
In the situation where the company is receiving fixed and paying floating, Bg, and Ba arecalculated in the same way, and equation (6.1) becomes
Vowap = Bay — Ba (6.3)
Example 6.2 Suppose that a financial institution pays six-month LIBOR and receives 8% perannum (with semiannual compounding) on a swap with a notional principal of $100 million and theremaining payment dates are in 3, 9, and 15 months. The swap has a remaining life of 15 months.The LIBOR rates with continuous compounding for 3-month, 9-month, and 15-month maturities are10%, 10.5%, and 11%, respectively. The 6-month LIBOR rate at the last payment date was 10.2%(with semiannual compounding). In this case, k = $4 million and k* = $5.1 million, so that
Bag = 4e OVS 4. de P-M0SH9/12 4 1 Qy OI TXIS/12= $98.24 million
Ba = 5. te 03/2 4 100670! *3/12= $102.51 million
Hence, the value of the swap is98.24 — 102.51 = —$4.27 million
If the bank had been in the opposite position of paying fixed and receiving floating, the value of theswap would be +3$4.27 million. A more precise calculation would take account of day countconventions and holiday calendars in determining exact cash flows and their timing.138 CHAPTER 6
 
Valuation in Terms of Forward Rate Agreements
We introduced forward rate agreements (FRAs) in Chapter 5. They are agreements that a certainpredetermined interest rate will apply to a certain principal for a certain period of time in the future.In Section 5.6 we showed that an FRA can be characterized as an agreement where interest at thepredetermined rate is exchanged for interest at the market rate of interest for the period in question.This shows that an interest rate swap is nothing more than a portfolio of forward rate agrecments.
Consider again the swap agreement between Intel and Microsoft in Figure 6.1. As illustrated inTable 6.1, this commits Microsoft to six cash flow exchanges. The first exchange is known at the timethe swap is negotiated. The other five exchanges can be regarded as FRAs. The exchange on March5, 2004, is an FRA where interest at 5% is exchanged for interest at the six-month rate observed in themarket on September 5, 2003; the exchange on September 5, 2004, is an FRA where interest at 5% isexchanged for interest at the six-month rate observed in the market on March 5, 2004; and so on.
As shown in Section 5.6, an FRA can be valued by assuming that forward interest rates arerealized. Since it is a portfolio of forward rate agreements, a plain vanilla interest rate swap can alsobe valued by assuming that forward interest rates are realized. The procedure is as follows:
. Calculate forward rates for each of the LIBOR rates that will determine swap cash flows.
Caleylate swa
Calculate swap cash flow that the LIBOR rates will equal ¢!
ming Couming tnat tne wi GUN Wi Gua
woh oe
. Set the swap value equal to the present value of these cash flows.
Example 6.3 Consider again the situation in the previous example. The cash flows that will beexchanged in 3 months have already been determined. A rate of 8% will be exchanged for 10.2%.The value of the exchange to the financial institution is
0.5 x 100 x (0.08 — 0.102)e °'*3/!? = —1.07To calculate the value of the exchange in 9 months, we must first calculate the forward ratecorresponding to the period between 3 and 9 months. From equation (5.1), this is
0.105 x 0.75 — 0.10 x 0.250.5or 10.75% with continuous compounding. From equation (3.4), this value becomes [1.044% withsemiannual compounding. The value of the FRA corresponding to the exchange in 9 months istherefore
= 0.1075
0.5 x 100 x (0.08 — 0.11044)e? O59? — 4 41
To calculate the value of the exchange in 15 months, we must first calculate the forward ratecorresponding to the period between 9 and 15 months. From equation (5.1), this isO.11 x 1.25 — 0.105 x 0.750.5or 11.75% with continuous compounding. From equation (3.4), this value becomes 12.102% with
semiannual compounding. The value of the FRA corresponding to the exchange in 15 months istherefore
= 0.1175
0.5 x 100 x (0.08 — 0.12102)e° B X17? = 4.79
The total value of the swap is—1.07 — 1.41 — 1.79 = —4.27
or --$4.27 million. This is in agreement with the calculation in Example 6.2 based on bond prices.Swaps 139
As already mentioned, the fixed rate in an interest rate swap is chosen so that the swap is worthzeto initially. This means that the sum of the value of the FRAs underlying the swap is zero. Itdoes not mean that the value of each individual FRA is zero. In general, some FRAs will havepositive values, whereas others will have negative values.
Consider the FRAs underlying the swap between the financial institution and Microsoft inFigure 6.4:
Value of FRA to financial institution < 0 when forward interest rate > 5.015%Value of FRA to financial institution = 0 when forward interest rate = 5.015%Value of FRA to financial institution > 0 when forward interest rate < 5.015%
Suppose that the term structure of interest rates is upward sloping at the time the swap is negotiated.This means that the forward interest rates increase as the maturity of the FRA increases. Because thesum of the values of the FRAs is zero, the forward interest rate must be less than 5.015% for theearly payment dates and greater than 5.015% for the later payment dates. The value to the financial
A Value of forwardcontract
InAil! woU
 
a)
A Value of forward
contractiil | | Maturity—>
(b)
 
Figure 6.8 Value of forward contracts underlying a swap as a function of maturity. In (a) the term
structure of interest rates is upward sloping and we receive fixed, or the term structure of interest rates is
downward sloping and we receive floating; in (b) the term structure is upward sloping and we receivefloating, or the term structure is downward sloping and we receive fixed140 CHAPTER 6
Dollars 8%
 
 
BritishPetroleum
 
Sterling 11%
Figure 6.9 A currency swap
institution of the FRAs corresponding to early payment dates is therefore positive, whereas the valueof the FRAs corresponding to later payment dates is negative. If the term structure is downwardsloping at the time the swap is negotiated, the reverse is true. The impact of the shape of the termstructure on the values of the forward contracts underlying a swap is summarized in Figure 6.8,
6.5 CURRENCY SWAPS
 
 
Another popular type of swap is known as a currency swap. In its simplest form, this involvesexchanging principal and interest payments in one currency for principal and interest payments inanother currency.
A currency swap agreement requires the principal to be specified in each of the two currencies.The principal amounts are usually exchanged at the beginning and at the end of the life of theswap. Usually the principal amounts are chosen to be approximately equivalent using the exchangerate at the swap’s initiation.
Mustration
Consider a hypothetical five-year currency swap agreement between IBM and British Petroleumentered into on February 1, 2001. We suppose that IBM pays a fixed rate of interest of 11% insterling and receives a fixed rate of interest of 8% in dollars from British Petroleum. Interest ratepayments are made once a year and the principal amounts are $15 million and £10 million. This istermed a fixed-for-fixed currency swap because the interest rate in both currencies is fixed. Theswap is shown in Figure 6.9. Initially, the principal amounts flow in the opposite direction to thearrows in Figure 6.9. The interest payments during the life of the swap and the final principalpayment flow in the same direction as the arrows. Thus, at the outset of the swap, IBM pays$15 million-and receives £10 million. Each year during the life of the swap contract, IBM receives$1.20 million (= 8% of $15 million) and pays £1.10 million (= 11% of £10 million), At the end ofthe life of the swap, it pays a principal of £10 million and receives a principal of $15 million. Thesecash flows are shown in Table 6.6.
Table 6.6 Cash flows to IBM in currency swap
Date Dollar cash flow (millions ) Sterling cash flow (millions)February 1, 2001 —15.00 +10.00February 1, 2002 +1,20 —1.10February 1, 2003 +120 —1.10February 1, 2004 11.20 ~ 1.10February 1, 2005 +1.20 —1.10
February 1, 2006 +16.20 —1t.10Swaps 141
 
Use of a Currency Swap to Transform Loans and Assets
A swap such as the one just considered can be used to transform borrowings in one currency toborrowings in another currency. Suppose that IBM can issue $15 million of U_S.-dollar-denominated bonds at 8% interest. The swap has the effect of transforming this transactioninto one where IBM has borrowed £10 million pounds at 11% interest. The initial exchange ofprincipal converts the proceeds of the bond issue from U.S. dollars to sterling. The subsequentexchanges in the swap have the effect of swapping the interest and principal payments fromdollars to sterling.
The swap can also be used to transform the nature of assets. Suppose that IBM can invest£10 million pounds in the U.K to yield 11% per annum for the next five years, but feels that theUS. dollar will strengthen against sterling and prefers a U.S.-denominated investment. The swaphas the effect of transforming the U.K. investment into a $15 million investment in the U.S.yielding 8%.
Comparative Advantage
Currency swaps can be motivated by comparative advantage. To illustrate this, we consideranother hypothetical example. Suppose the five-year fixed-rate borrowing costs to GeneralMotors and Qantas Airways in U.S. dollars (USD) and Australian dollars (AUD) are as shownin Table 6.7. The data in the table suggest that Australian rates are higher than U.S. interestrates. Also, General Motors is more creditworthy than Qantas Airways, because it is offered amore favorable rate of interest in both currencies. From the viewpoint of a swap trader, theinteresting aspect of Table 6.7 is that the spreads between the rates paid by General Motors andQantas Airways in the two markets are not the same. Qantas Airways pays 2% more thanGeneral Motors in the U.S. dollar market and only 0.4% more than General Motors in theAUD market.
This situation is analogous to that in Table 6.4. Gencral Motors has a comparative advantage inthe USD market, whereas Qantas Airways has a comparative advantage in the AUD market. InTable 6.4, where a plain vanilla interest rate swap was considered, we argued that comparativeadvantages were largely illusory. Here we are comparing the rates offered in two differentcurrencies, and it is more likely that the comparative advantages are genuine. One possible sourceof comparative advantage is tax. General Motors’s position might be such that USD borrowingslead to lower taxes on its worldwide income than AUD borrowings. Qantas Airways’s positionmight be the reverse. (Note that we assumc that the interest rates in Table 6.7 have been adjusted toreflect these types of tax advantages.)
We suppose that General Motors wants to borrow 20 million AUD and Qantas Airways wants
Table 6.7 Borrowing rates providing basisfor currency swap
 
 
USD* AUD*General Motors 5.0% 12.6%Qantas Airways 7.0% 13.0%
* Quoted rates have been adjusted to reflect thedifferential impact of taxes.142 CHAPTER 6
 
USD 5.0%
       
USD 6.3%
    
GeneralMotors
Financial
Qantasinstitution
<«—_— Aways | AUD 13.0%
USD 5.0%
      
 
 
 
 
AUD 11.9% AUD 13.0%
Figure 6.10 A currency swap motivated by comparative advantage
to borrow 12 million USD and that the current exchange rate is 0.6 USD/AUD. This creates aperfect situation for a currency swap. General Motors and Qantas Airways each borrow in themarket where they have a comparative advantage; that is, General Motors borrows USD whereasQantas Airways borrows AUD. They then use a currency swap to transform General Motors’sloan into an AUD loan and Qantas Airways’s loan into a USD loan.
As already mentioned, the difference between the dollar interest rates is 2%, whereas thedifference between the AUD interest rates is 0.4%. By analogy with the interest rate swap case,we expect the total gain to all parties to be 2.0 — 0.4 = 1.6% per annum.
There are many ways in which the swap can be organized. Figure 6.10 shows one way swapsmight be entered into with a financial institution. General Motors borrows USD and QantasAirways borrows AUD. The effect of the swap is to transform the USD interest rate of 5% perannum to an AUD interest rate of 11.9% per annum for General Motors. As a result, GeneralMotors is 0.7% per annum better off than it would be if it went directly to AUD markets.Similarly, Qantas exchanges an AUD loan at 13% per annum for a USD loan at 6.3% perannum and ends up 0.7% per annum better off than it would be if it went directly to USDmarkets. The financial institution gains 1.3% per annum on its USD cash flows and loses 1.1%per annum on its AUD flows. If we ignore the difference between the two currencies, the financialinstitution makes a net gain of 0.2% per annum. As predicted, the total gain to all parties is1.6% per annum.
Each year the financial institution makes a gain of USD 156,000 (= 1.3% of 12 million) andincurs a loss of AUD 220,000 (= 1.1% of 20 million). The financial institution can avoid anyforeign exchange risk by buying AUD 220,000 per annum in the forward market for each year ofthe life of the swap, thus locking in a net gain in USD.
It is possible to redesign the swap so that the financial institution makes a 0.2% spread in USD.Figures 6.11 and 6.12 present two alternatives. These alternatives are unlikely to be used in practice
 
    
      
 
 
 
 
USD 5.0% USD 5.2%General [*————__ Financial QantasMotors [+ _ institution AirwaysUSD 5.0% AUD 11.9% AUD 11.9% AUD 13.0%
 
 
Figure 6.11 Alternative arrangement for currency swap:Qantas Airways bears some foreign exchange risk
       
      
      
USD 6.1% USD 6.3%
           
Financialinstitution
GeneralMotors
QantasAirways
      
AUD 13.0%
AUD 13.0% AUD 13.0%
 
 
Figure 6.12 Alternative arrangement for currency swap:General Motors bears some foreign exchange riskSwaps 143
because they do not lead to General Motors and Qantas being free of foreign exchange risk.> InFigure 6.11, Qantas bears some foreign exchange risk because it pays 1.1% per annum in AUDand 5.2% per annum in USD. In Figure 6.12, General Motors bears some foreign exchange riskbecause it receives 1.1% per annum in USD and pays 13% per annum in AUD.
6.6 VALUATION OF CURRENCY SWAPS
 
 
In the absence of default risk, a currency swap can be decomposed into a position in two bonds, asis the case with an interest rate swap. Consider the position of IBM in Table 6.6 some time after theinitial exchange of principal. It is short a GBP bond that pays interest at 11% per annum and longa USD bond that pays interest at 8% per annum.In general, if we define Vyap as the value in U.S. dollars of a swap where dollars are received anda foreign currency is paid, thenVowap = Bp — SoBr
where By; is the value, measured in the foreign currency, of the foreign-denominated bondunderlying the swap, Bp is the value of the U.S. dollar bond underlying the swap, and Sy is thespot exchange rate (expressed as number of units of domestic currency per unit of foreigncurrency). The value of a swap can therefore be determined from LIBOR rates in the twocurrencies, the term structure of interest rates in the domestic currency, and the spot exchangerate. Similarly, the value of a swap where the foreign currency is received and sterling is paid is
Vowap = SoBr -Bp
Example 6.4 Suppose that the term structure of interest rates is flat in both Japan and the UnitedStates. The Japanese rate is 4% per annum and the U.S. rate is 9% per annum (both with continuouscompounding). A financial institution has entered into a currency swap in which it receives 5% perannum in yen and pays 8% per annum in dollars once a year. The principals in the two currencies are$10 million and 1,200 million yen. The swap will last for another three years, and the currentexchange rate is 110 yen = $1. In this case,
Bp =0 Be 0.09x1 + 0.827 0-092 +10 Be 0.093= 9.644 million dollars
Bp = 60e7°**! 4. 609942 4 126007043= 1,230.55 million yen
1,230.55
Tie 9.644 = 1.543 million
If the financial institution had been paying yen and receiving dollars, the value of the swap wouldhave been —$1.543 million.
5 Usually it makes sense for the financial institution to bear the foreign exchange risk, because it is in the bestposition to hedge the risk.144 CHAPTER 6
Decomposition into Forward Contracts
An alternative decomposition of the currency swap is into a series of forward contracts. Consideragain the situation in Table 6.6. On each payment date IBM has agreed to exchange an inflow of$1.2 million and an outflow of £1.1 million. In addition, at the final payment date, it has agreed toexchange a $15 million inflow for a £10 million outflow. Each of these exchanges represents aforward contract. In Section 3.8 we saw that forward contracts can be valued on the assumptionthat the forward price of the underlying asset is realized. This provides a convenient way of valuingthe forward contracts underlying a currency swap.
Example 6.5 Consider the situation in the previous example. The current spot rate is 110 yen perdollar, or 0.009091 dollar per yen. Because the difference between the dollar and yen interest rates is5% per annum, equation (3.13) can be used to give the one-year, two-year, and three-year forwardexchange rates as
0.0090912°°*! = 0.009557
0.00909 1e%"5*? = 0.0100470.00909 129°? = 0.010562
respectively. The exchange of interest involves receiving 60 million yen and paying $0.8 million. Therisk-free interest rate in dollars is 9% per annum. From equation (3.8), the values of the forwardcontracts corresponding to the exchange of interest are (in millions of dollars)
(60 x 0,009557 — 0.8)e~°*! — —0.2071(60 x 0.010047 — 0.8)e~°*? = —0.1647(60 x 0.010562 — 0.8)e°*3 = —0.1269
The final exchange of principal involves receiving 1,200 million yen and paying $10 million. Fromequation (3.8), the value of the forward contract corresponding to the exchange is (in millions ofdollars)
(1,200 x 0.010562 — 10)e7° 3 = 2.0416
The total value of the swap is 2.0416 — 0.1269 — 0.1647 — 0.2071 = $1.543 million, which is inagreement with the result of the calculations in the previous cxample.
The value of a currency swap is normally zero when it is first negotiated. If the two principals areworth exactly the same using the exchange rate at the start of the swap, the value of the swap isalso zero immediately after the initial exchange of principal. However, as in the case of interest rateswaps, this does not mean that each of the individual forward contracts underlying the swap haszero value. It can be shown that when interest rates in two currencies are significantly different, thepayer of the low-interest-rate currency is in the position where the forward contracts correspondingto the early exchanges of cash flows have positive values, and the forward contract correspondingto final exchange of principals has a negative expected value. The payer of the high-interest-ratecurrency is likely to be in the opposite position; that is, the early exchanges of cash flows havenegative values and the final exchange has a positive expected value.
For the payer of the low-interest-rate currency, the swap will tend to have a negative value duringmost of its life. The forward contracts corresponding to the early exchanges of payments havepositive values, and, once these exchanges have taken place, there is a tendency for the remainingforward contracts to have, in total, a negative value. For the payer of the high-interest-rateSwaps 145
currency, the reverse is true. The value of the swap will tend to be positive during most of its life.These results are important when the credit risk in the swap is being evaluated.
6.7 CREDIT RISK
 
Contracts such as swaps that are private arrangements between two companies entail credit risks.Consider a financial institution that has entered into offsetting contracts with two companies (seeFigure 6.4, 6.5, or 6.7). If neither party defaults, the financial institution remains fully hedged.A decl