Currency and Interest Rate Swaps

TTh 3:15-4:30 Gates B01 Final Exam MS&E 247S Friday Aug 14 2009 12:15PM-3:15PM Gates B01 (alternate arrangement???) Or Saturday Aug 15 2009 7PM-10PM Gates B01 (official date and time???) Remote SCPD participants will also take the exam on Friday, 8/14. Please Submit Exam Proctor s Name, Contact info as SCPD requires, also c.c. to ffuy@stanford.edu, preferably a week before the exam. Local SCPD students please come to Stanford to take the exam. Light refreshments will be served. Handout #16 Derivative Security Markets Currency and Interest Rate Swaps http://stanford2009.pageout.net

Reading Assignments for this Week Scan Read Levich Chap 13 Pages Currency and Interest Rate Swaps Luenberger Chap Pages Solnik Chap Pages McDonald Ch 8 Swaps Pages 219-246 Fundamentals of Derivative Markets Wooldridge Chap Pages 13-2

Derivative Security Markets Currency and Interest Rate Swaps MS&E 247S International Investments Yee-Tien Fu

Medical Swap vs. Financial Swap http://www.pageout.net/user/www/s/t/stanford2007/medical%20swap.pdf 13-4

Mileage Swap vs. Financial Swap Frequent flier programs have been around for nearly three decades and billions of miles go unused. Airlines used to prohibit swaps of frequent-flier miles - it's still in the fine print of many loyalty programs. But now some are perfectly fine with exchanges like the one that Hintz made - they collect a fee on every trade. NEXT PAGE 13-5

For some fliers, trading miles is the way to go By DAVID KOENIG, AP Airlines Writer DALLAS (AP) -- Scott Hintz needed more miles with American Airlines to book a free trip to Morocco this spring, and he had several thousand miles from another carrier that he thought might be just the ticket. The San Francisco travel executive went online, found a willing trader for his Alaska Airlines miles and made a swap. In May he was roaming North Africa. "I took miles out of some programs I don't use and got some value out of them," says Hintz, who calls himself "a miles junkie." 13-6

Introduction to Swaps A capital market swap represents an agreement to exchange cash flows between two parties, usually referred to as counterparties. A swap agreement commits each counterparty to exchange an amount of funds, determined by a formula, at regular intervals until the swap expires. In the case of a currency swap, there is an initial exchange of currency and a reverse exchange at maturity. 13-7

Introduction to Swaps Like futures and options, a swap is a derivative security. A swap is equivalent to a collection of forward contracts that call for an exchange of funds at specified times in the future. Like forward contracts, a swap can be used to speculate, to hedge an exposure, or to replicate another security in an effort to enhance investment returns or to lower borrowing costs. 13-8

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Introduction to Swaps Since a swap can be replicated using forward contracts, why does the swap market exist, and why has it grown so popular? A swap reduces transaction costs by allowing the counterparties to combine many transactions (forward contracts) into one (the swap). In addition, the legal structure of a swap transaction may have advantages that reduce the risk to each party in the event of a default by the other party. 13-10

Introduction to Swaps The cash flows of a swap were linked: if firm A could not pay firm B, then firm B felt excused from having to pay firm A. Whether swaps always reflect this right-ofoffset is a critical point. In addition, as a new financial product, the currency swap was not covered by any accounting disclosure or security registration requirements. 13-11

Introduction to Swaps We will focus primarily on the economic fundamentals and financial characteristics of basic interest rate and currency swap agreements. 13-12

Structure of a Back-to-Back on Parallel Loan Basic Swap In the United Kingdom In the Netherlands British parent firm Direct loan in pounds Indirect financing Dutch parent firm Direct loan in guilders Dutch firm s affiliate in the United Kingdom British firm s affiliate in the Netherlands Figure 13.1 Pg 448 13-13

The Role of Capital Controls Suppose a Brazilian affiliate wants to transfer funds to its U.S. parent firm beyond what it was allowed to repatriate home. It could effectively do so if it made a loan to a Brazilian affiliate of a French firm and the French parent simultaneously lent funds to the U.S. parent. Since the legal barrier imposes a cost, the U.S firm is willing to provide a financial incentive to the French firm to take part in the deal. 13-14

Structure of a Back-to-Back on Parallel Loan Variation In the United States In Brazil French parent firm Direct loan in dollars Indirect financing U.S. firm s affiliate in Brazil Direct loan in cruzados U.S. parent firm French firm s affiliate in Brazil Figure 13.1 Pg 448 13-15

The Role of Capital Controls Though useful, back-to-back and parallel loans present certain drawbacks : 1 Identifying a counterparty is both time consuming and costly. 2 Legally, the two loans are separate and distinct. Hence, one party s default will not release the other party from its commitments under the other loan. 3 For accounting and regulatory purposes, the loans are loans. Thus, the firm s borrowing capacity, credit rating etc can be affected. 13-16

Factors Favoring the Risk of Swaps The currency swap evolved as a way to simplify and speed up the exchange of currency cash flows between counterparties. In addition, it linked the two cash flows : Only the net difference between the two cash flows is paid. If A cannot pay B, then B may be excused from paying A. As a new financial product, it was not covered by any accounting disclosure or security registration requirements. 13-17

The Swap Market The notional value of outstanding swaps is the underlying amount on which swap payments are based. A more meaningful indicator of the economic significance of outstanding swaps is the gross market value, which reflects the cost that one party would pay to replace a swap at market prices in the event of a default. Gross market value represents the gross exposure associated with swap contracts. 13-18

The Basic Cash Flows of a Currency Swap Firms A and B can each issue a 7-year bond in either the US$ or SFr market. Firm A has a comparative advantage in borrowing US$ while firm B has a comparative advantage in borrowing SFr. By borrowing in their comparative advantage currencies and then swapping, lower cost Figure 13.2 Pg 453 US$ finance SFr finance financing is possible. Firm A 10% Firm B 11.5% 5% 6% Difference (A-B) -1.5% -1.0% -0.5% 13-19

The Basic Cash Flows of a Currency Swap $ at t 0 SFr at t 0 A Borrows $ at 10% 10.75% (US$) 5.5% (SFr) $ at t 7 SFr at t 7 B Borrows SFr at 6% Together, A and B save 0.5%. Note that if a bank or swap dealer intermediates the transaction and charges a fee, the aggregate interest savings will be reduced. Figure 13.2 Pg 453 13-20

The Basic Cash Flows of a Currency Swap: Result of Strategy Firm A pays 5.5% (to B) on its SFr150 million loan. But firm A also pays 10.0% interest on its US$ bonds while receiving 10.75% interest on its US$100 million loan to B -- or a net inflow of 0.75%. Thus, A pays (approximately) 4.75% net interest on its SFr loan. This represents a 0.25% savings in relation to its own cost of borrowing SFr. Note that the calculation is approximate because 1% interest on US$ is not precisely the same as 1% interest on SFr. 13-21

The Basic Cash Flows of a Currency Swap: Result of Strategy Firm B pays 10.75% (to A) on its US$100 million loan. But B also pays 6.0% interest on its SFr bonds and receives 5.5% interest on its SFr 150 million loan to A -- or a net outflow of 0.5%. Thus, B pays (approximately) 11.25% net interest on its US$ loan. This represents a 0.25% savings in relation to its own cost of borrowing US$. Note that the calculation is approximate because 1% interest on US$ is not precisely the same as 1% interest on SFr. 13-22

A Summary of the IBM / World Bank Currency Swap IBM and the World Bank can each issue a 7- year bond in either the US$ or SFr market. The World Bank had an absolute advantage in the US$ market, while IBM had an absolute advantage in the SFr bond market. Examining these borrowing costs, we see that the firms could save 25bp by entering Box 13.1 Pg 455 $ finance SFr finance into a currency swap. IBM U.S. Treasury + 45 bp Swiss Treasury + 0 bp World Bank U.S. Treasury + 40 bp Swiss Treasury + 20 bp Difference 5bp - 20bp 25bp 13-23

A Summary of the IBM / World Bank Currency Swap $ at t 0 SFr at t 0 World Bank Borrows $ at U.S. Treasury + 40 U.S. Treas.+40 Swiss Treas.+10 $ at t 7 SFr at t 7 IBM Borrows SFr at Swiss Treasury + 0 The total cost for IBM was (Swiss T. + 0bp) + (U.S. T. + 40bp) - (Swiss T. + 10bp) = U.S. T. + 30bp. So IBM saved 15bp. The total cost for the World Bank was (U.S. T. + 40bp) + (Swiss T. + 10bp) - (U.S. T. + 40bp) = Swiss T. + 10bp. So, the Bank saved 10bp. Box 13.1 Pg 455 13-24

Saturation and scarcity value In 1981, both IBM and the World Bank were rated AAA for credit risk. But in the real world, all AAA credits are not necessarily awarded the same interest cost of funds, even if the bond issues share similar characteristics. The reasoning is based, in part, on saturation and scarcity value. Other things being equal, investors seeking a portfolio of AAA bonds prefer to hold bonds from a broad set of issuers in order to diversify the idiosyncratic risks of any single issuer. An issuer who has not saturated the market may enjoy a scarcity value and be able to issue bonds at a lower rate. This effect is more likely among AAA-rated issuers, given the small universe of AAA-rated issuers. 13-25

Saturation and scarcity value Until 1981, the World Bank was a frequent issuer of bonds in the Swiss market in order to capture the low nominal interest rate in SFr. With the demand for World Bank bonds saturated at prevailing rates, Swiss investors demanded a higher interest rate to hold additional World Bank bonds. IBM, on the other hand, viewed themselves as a US$-based firm and borrowed exclusively in the US$ bond markets. Swiss investors were willing to pay a premium (reflecting a scarcity value) to bring IBM as a new AAA-issuer into their portfolios. 13-26

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Interest Rate Swap Fixed-Rate Payer Floating-Rate Payer Pays fixed rate in the swap Pays floating rate in the swap Receives floating in the swap Receives fixed in the swap Has bought a swap Has sold a swap Is long a swap Is short a swap Is short the bond market Is long the bond market 13-28

Interest Rate Swap In Figure 13.3, we show two firms, A and B, that can issue a US$ denominated bond in either fixed-rate or floating-rate terms. The annual interest costs are assumed to be 9.0% and 10.5% respectively, for A and B in the fixed-rate bond market. In addition, each firm can arrange floating rate financing, perhaps through bank lending or a commercial paper (CP) program. We assume that firm A pays six-month LIBOR plus zero basis points, while firm B pays six-month LIBOR plus 50 basis points. This example assumes that interest is paid semiannually and the floating interest rate is reset every six months. 13-29

The Basic Cash Flows of an Interest Rate Swap Firm A Firm B Difference (A-B) Fixedrate finance 9% 10.5% -1.5% Floatingrate finance LIBOR +0.0% LIBOR +0.5% -0.5% -1.0% A Borrows at 9.0% fixed 9.75% LIBOR +.25 B Borrows at LIBOR + 0.50% floating Figure 13.3 Pg 456 13-30

The Basic Cash Flows of an Interest Rate Swap To explain the transactions in an interest rate swap, assume that in period t 0, firm A issues a seven-year bond for $100 million at a fixed rate of 9% and B obtains bank financing for $100 million at a floating rate equal to six-month LIBOR + 0.5%. In our example, the principal amounts are identical, so there is no need to actually exchange principal as in the currency swap example. However (and as if there were an exchange of principal), A agrees to pay LIBOR + 0.25% interest on $100 million to B, while B agrees to pay 9.75% interest on $100 million to A. 13-31

The Basic Cash Flows of an Interest Rate Swap In years t 1 until t 7, firms A and B make interest payments to each other as stipulated in the swap agreement, plus paying interest on the original bonds they have issued. At time t 7, the swap contract matures. A and B make their final interest payments to each other, A retires its outstanding bond issue, and B pays off its bank loan. 13-32

The Basic Cash Flows of an Interest Rate Swap What is the result of this strategy? Firm A pays LIBOR + 0.25% interest (to B) and 9.0% on its fixed-rate bonds, while receiving 9.75% interest from B -- or a net interest cost of LIBOR - 0.50%. Thus, A saves 0.50% in relation to its own cost of floating-rate funds. 13-33

How is the value of a swap determined? Once the swap transaction is completed, changes in market interest rates will change the payments of the floating-rate side of the swap. The value of an interest-rate swap is the difference between the present value of the payments of the two sides of the swap. The three-month LIBOR forward rates from the current Eurodollar CD futures contracts are used to (i) calculate the floating-rate payments and (ii) determine the discount factors at which to calculate the present value of the payments. 13-34

What factors affect the swap rate? For the swap rate, we have: swap rate = N t= 1 present value of floating-rate payments days in period t notional amount forward discount factor for period 360. From this equation, we see that the swap rate is determined by the present value of floating-rate payments, the notional amount, the number of periods in a year, and the forward discount factor. t 13-35

The present value of floating-rate payments is also determined by the notional amount, the number of periods in a year, and the forward discount factor. It is also determined by the reference rate (such as LIBOR) and forward rates based on this benchmark. It is important to emphasize that the reference rate at the beginning of period t determines the floating rate that will be paid for the period. However, the floatingrate payment is not made until the end of period t. We should also point out that the same forward rates that are used to compute the floating-rate payments those obtained from the Eurodollar CD futures contract are used in computing the forward discount factors for each period t. 13-36

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Profit and Loss from Entering into an Interest Rate Swap When Interest Rates Are Variable Behavior of Floating Interest Rates Interest Rates Rise Relative to Expectations Interest Rates Fall Relative to Expectations Speculative position Pay floating and receive fixed Speculative loss, swap has negative value, out-of-themoney swap Speculative gain, swap has position value, in-themoney swap Pay fixed and receive floating Speculative gain, swap has position value, in-themoney swap Speculative loss, swap has negative value, out-of-the-money swap Table 13.4 Pg 464 13-40

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Behavior of Short-Term Interest Rates and the Valuation of Fixed Floating Swap Valuation Effects for Paying Fixed and Receiving Floating Period 1 Period 2 Period 3 Period 4 Period 5 Initial Euro-$ Interest Rate i 1 = 5.22 Positive value, In the money i 5,5 = 5.42 i 5,4 = 5.32 i 5,3 = 5.22 i 5,2 = 5.12 Negative value, Out of the money i 5,1 = 5.02 Figure 13.5 Pg 465 13-43

The Amortization Effect and the Diffusion Effect in a Long-Term Interest Rate Swap Potential Exposure (%) Amortization Effect Time Diffusion Effect Figure 13.6A Pg 466 13-44

The Overall Risk in a Long-Term Interest Rate Swap Potential Exposure (%) Interaction of Amortization and Diffusion Effect Time Figure 13.6B Pg 466 13-45

Expected Credit Exposures on Interest Rate Swaps: The Maturity Effect Percent of Notional Principal 10 8 6 4 2 0 Figure 13.7A Pg 467 1-year 3-year 5-year 7-year 10-year 4 8 12 16 20 Semiannual Periods 13-46

Expected Credit Exposures on 10-Year Interest Rate Swaps: The Interest Rate Level Effect Percent of Notional Principal 10 8 6 4 2 0 Figure 13.7B Pg 468 13% 9% 7% 11% 4 8 12 16 20 Semiannual Periods 13-47

Swaps & Linkages Across International Capital Markets Currency of Denomination Examples A Dollar-denominated straight Eurobond B Eurodollar floatingrate note (FRN) C Samurai bond D Euroyen floatingrate note (FRN) Figure 13.8 Pg 469 Interest Rate Base Currency X Currency Y Fixed Rate Floating Rate Asset or Liability Asset or Liability A Interest Rate Swap B Fixed-Fixed Currency Swap C Cross Currency Cross Currency Interest Rate Swap Interest Rate Swap Interest Rate Swap Floating-Floating Currency Swap D 13-48

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An Example of Price Quotations in the Swap Market Table 13.5 presents a sample of swap quotations from a major dealer. Various interest rate swap quotations within the US$ segment are shown in panel A of Table 13.5. Various cross-currency interest rate swap quotations for the US$ against other currencies are shown in panel B of Table 13.5. A diagram illustrating how the quotation apply to the dealer and the counterparties is in panel C. 13-50

An Example of Price Quotations in the Swap Market Panel A: U.S. Dollar Interest Rate Swaps Maturity Treasury Yield Treasury vs. LIBOR 2 5.94 Bid Offer 18 20 Treasury vs. T-Bills Treasury vs. CP Bid Offer Bid Offer -21-16 12 16 Note: Quotes are in basis points over/under Treasury bond yield. Table 13.5 Pg 470 13-51

An Example of Price Quotations in the Swap Market Panel B: Non-U.S. Dollar Interest Rate Swaps Maturity Japanese Yen Pound Sterling 2 Bid Offer Bid Offer 1.49 1.53 6.507 6.557 Deutsche Mark Swiss Franc Bid Offer Bid Offer 4.035 4.085 2.990 3.090 Note: Quotes are on an actual/365 day semi-annual basis. Table 13.5 Pg 470 13-52

Price Quoting Conventions in the Swap Market Bid Quote Swap Quotes Offer Quote Counter Party A Swap dealer pays fixed rate Swap Dealer Swap dealer receives fixed rate Counter Party B Swap dealer receives floating rate Swap dealer pays floating rate Quotes are given from the perspective of the swap dealer. The convention is to quote only the fixed side of the swap. All fixed quotes are against LIBOR unless otherwise stated. Panel C of Table 13.5 Pg 470 13-53

Constructing a Fixed-Fixed Currency Swap Suppose a US firm (A) issues a seven-year Euro-Y straight bond with a coupon of 3.80%, and that a Japanese firm (B) issues a seven-year Euro-$ straight bond with a coupon of 7.40%. Assume that A wishes to obtain fixed-rate US$ financing and that B wishes to obtain fixed-rate Y financing and that both are willing to trade at the quotes in Table 13.5 from the swap dealer at Merrill Lynch. The relevant prices to use will be the seven-year T-bond versus LIBOR quotes for US$ interest rate swaps in panel A, and the Japanese yen cross-currency swap in panel B. 13-54

Construction of a Fixed-Fixed Currency Swap 3.10%( ) 3.14%( ) U.S. Firm A LIBOR ($) LIBOR ($) Merrill Lynch LIBOR ($) LIBOR ($) Japanese Firm B 6.67%($) 6.65%($) Borrows fixed-rate bond at 3.80% Borrows fixed-rate $ bond at 7.40% Table 13.6 Pg 472 13-55

Construction of a Fixed-Fixed Currency Swap To convert its fixed-rate Euro-Y bond into a fixed-rate US$ liability, firm A enters into two swaps with Merrill Lynch: 1. A cross-currency swap paying $-LIBOR and receiving 3.10% in Y. 2. An interest rate swap paying 6.67% in $ (equal to 6.32% T-bond rate + 0.35%) and receiving $- LIBOR. We can see that the two LIBOR portions cancel, leaving firm A with a fixed-rate US$ liability costing 7.37%. 13-56

Construction of a Fixed-Fixed Currency Swap Firm B also enters into two swaps with Merrill Lynch: 1. A cross-currency swap receiving $-LIBOR and paying 3.14% in Y. 2. An interest rate swap receiving 6.65% in $ (equal to 6.32% T-bond rate + 0.33%) and paying $-LIBOR. Again, we can see that the two LIBOR portions cancel, leaving firm B with a fixed-rate Y liability costing 3.89%. 13-57

Construction of a Fixed-Fixed Currency Swap As the intermediary in this transaction, Merrill Lynch earns 0.04% in Y and another 0.02% in US$ on a per annum basis over the seven-year life of the swap. In addition, Merrill Lynch receives a fee for originating each swap. All of these transactions are summarized in Table 13.6. 13-58

Applications of Swaps: Magnifying Risk and Return Many of the illustrations in this chapter have linked a swap with a bond issue, but these decisions are separable. A firm can issue a bond in one year and then decide to swap later, using the swap as a risk management tool. However, a firm could enter into a swap without a prior bond issue. This transaction is the same as a pure speculation on the direction of exchange rates or interest rates. 13-59

Applications of Swaps: Magnifying Risk and Return The swaps discussed in the chapter could be termed plain vanilla as the payoffs are governed by the simple differential between two specific interest rates. But more exotic swaps could be designed, with payoffs proportional to twice the interest differential, or the square of the interest differential. In principle, these exotic contracts could reduce the firm s exposure to risk from its core business activities. But it is also true that exotic swaps are a way to enhance speculative return and risk, if these contracts are not tempered with other hedging transactions. 13-60

An Unsuccessful Exotic Swap Procter and Gamble (P&G) (based in Cincinnati and with $30 billion in annual sales) lost $157 million on an exotic swap whose payments ( in most cases ) were defined by the formula: 17.0415 x (5-year Treasury rate) - (price of 6.25 percent Treasury due 8/2023) -0.75% The amount of interest that P&G would pay under this formula is shown in Table 13.7. 13-61

Table 13.7 Interest Cost (Premium over the CP Rate) in the Procter & Gamble/Bankers Trust Interest Rate Swap 30-Year Interest Rate 5-year Int 6% 7% 8% 5% -0.75% -0.75% 4.20% 6% -0.75% 10.80% 21.20% 7% 15.10% 24.90% 38.20% 13-62

An Unsuccessful Exotic Swap This arrangement would have reduced P&G s funding costs below the commercial paper rate if short-term interest rates fell. But if interest rates rose, P&G was subject to enormous borrowing costs on its $200 million notional value. P&G closed its swap position to cap their loss, and filed suit against the swap dealer (Bankers Trust), alleging that the dealer failed to make sufficient disclosures of the risks involved in the transaction. Bankers Trust claimed that it had acted in good faith and that it was dealing with a sophisticated investor with extensive experience in exotic derivatives. 13-63

An Unsuccessful Exotic Swap P&G added a second money-losing DM interest rate swap to its lawsuit, bringing its total claim against Bankers Trust to $200 million. P&G and Bankers Trust settled their dispute in May 1996, after Bankers Trust agreed to absorb at least $150 million of P&G s loss. Around this time, it was reported that some banks had chosen to absorb losses on swap transactions, rather than risk bad publicity or litigation. 13-64

Assignment from Chapter 13 Exercises 1, 2. 13-65

1. Suppose Firm ABC can issue 7-year bonds in the US at the fixed rate of 8% and in France at 13%. Suppose Firm XYZ can issue 7- year bonds at the fixed rate of 10% in the US in US$ and at 14% in France in FFr. a. Which firm has a comparative advantage in the French capital market? b. How would you advise both firms so that they take advantage of each other's comparative advantage in the US and French capital markets? c. How much could be saved in borrowing costs by both firms? d. What could cause the relative comparative advantages in international credit markets? 13-66

SOLUTION: a. ABC XYZ Difference US$ 8% 10% -2% FFr 13% 14% -1% -1% XYZ has a comparative advantage in the French franc market; ABC has a comparative advantage in the US$ market. b. Each firm has a comparative advantage in different markets. They should take advantage of that edge, then swap the proceeds, thus realizing borrowing cost savings. 13-67

c. Total Costs: ABC Pays 8.0% XYZ Pays 14.0% Pays 13.5% Pays 9.0% Receives 9.0% Receives 13.5% Net 12.5% Net 9.5% Savings.5% Savings.5% Total Savings: 1% d. Different comparative advantage for both firms may arise because a firm's local credit market is saturated with the firm s debt and would place value in the availability of debt issues by a foreign firm. Different valuation on the same credit instrument could also arise because the French and US credit markets make different assessments of the riskiness of the same firms. 13-68

2. Suppose two parties enter a 5-year interest rate swap to exchange one-year LIBOR plus 50 basis points (bp) for a fixed rate on $100 million notional principal. a. If LIBOR turns out to be 10% in year 1, 9% in year 2, 9% in year 3, 8% in year 4 and 8.5% in year 5, what cash flows will be exchanged between the two parties? Assume a flat Eurodollar yield curve at 10%. b. What is the value of the swap? c. What fixed rate in the swap agreement will make the value of the swap equal to zero? 13-69

SOLUTIONS: a. The cash-flow pattern is as follows: 1 2 3 4 5 Fixed 10% 10% 10% 10% 10% Cash-Flows 10 10 10 10 10 LIBOR +50 bp 10.5% 9.5% 9.5% 8% 8.5% Cash-Flows 10.5 9.5 9.5 8 8.5 Difference -.5.5.5 2 1.5 b. The NPV at 10% yields a positive value of $2.63 for the fixedrate payer. c. A fixed-rate of approximately 9.375% will make the NPV equal to zero. 13-70

Interest Rate Swap (from Fabozzi: Bond Markets, Analysis and Strategies) 14. Consider the following interest-rate swap: the swap starts today, January 1 of year 1 (swap settlement date) the floating-rate payments are made quarterly based on actual / 360 the reference rate is three-month LIBOR the notional amount of the swap is $40 million the term of the swap is three years Answer the following questions. (a) Suppose that today s three-month LIBOR is 5.7%. What will the fixed-rate payer for this interest rate swap receive on March 31 of year 1 (assuming that year 1 is not a leap year)? The quarterly floating-rate payments are based on an actual or 360-day count convention. This convention means that 360 days are assumed in a year, and that in computing the interest for the quarter the actual number of days in the quarter is used. The floating-rate payment is set at the beginning of the quarter but paid at the end of the quarter that is, the floating-rate payments are made in arrears. For our problem, today s three-month LIBOR is 5.7%. Thus, the fixed-rate payer receives payment based on this rate on March 31 of year 1 the date when the first quarterly swap payment is made. There is no uncertainty about what this floating-rate payment will be. In general, the floating-rate payment is given as: floating-rate payment = notional amount three-month LIBOR number of days in period 360. In our problem, assuming a non-leap year, the number of days from January 1 of year 1 to March 31 of year 1 (the first quarter) is 90. If three-month LIBOR is 5.7%, then the fixed-rate payer will receive a floating-rate payment on March 31 of year 1 as shown below: 1

floating-rate payment = $40,000,000 0.057 90 360 = $570,000. (b) Assume the Eurodollar CD futures price for the next seven quarters is as follows: Quarter Starts Quarter Ends Number of Days in Quarter Eurodollar CD Futures Price April 1 year 1 June 30 year 1 91 94.10 July 1 year 1 Sept 30 year 1 92 94.00 Oct 1 year 1 Dec 31 year 1 92 93.70 Jan 1 year 2 Mar 31 year 2 90 93.60 April 1 year 2 June 30 year 2 91 93.50 July 1 year 2 Sept 30 year 2 92 93.20 Oct 1 year 2 Dec 31 year 2 92 93.00 Compute the forward rate for each quarter. Forward rate for period 1 is: three-month LIBOR today number of days in period 360. Inserting our values, we have: 5.70% 90 360 = 1.425%. The forward rate for periods 2 through 8 is given by: forward rate = 100 Eurodollar CD Futures price for that period 100 number of days in period 360. Inserting the value for period two (i.e., April 1 year1 to June 30 year 1), we have: forward rate = 100 94.10 100 91 360 = 0.0149138 or 1.4913889%. 2

Similarly, for periods 3, 4, 5, 6, 7, and 8, the respective forward rates are: 1.5333333%, 1.6100%, 1.6000%, 1.6430556%, 1.7377778%, and 1.7888889%. (c) What is the floating-rate payment at the end of each quarter for this interest-rate swap? The floating-rate payment for each period is the forward rate for that period, as given in the part (b), times the notional amount of $40 million. For period 1, we have forward rate for period 1 notional amount = 0.01425 $40,000,000 = $570,000.00. For period 2, we have: forward rate for period 2 notional amount = 0.014913889 $40,000,000 = $596,555.56. Similarly, for periods 3, 4, 5, 6, 7, and 8, the respective forward rates are: $613,333.33, $644,000.00, $640,000.00, $657,222.22, $695,111.11, and $715,555.56. 15. Answer the following questions. (a) Assume that the swap rate for an interest-rate swap is 7% and that the fixed-rate swap payments are made quarterly on an actual or 360-day basis. If the notional amount of a two-year swap is $20 million, what is the fixed-rate payment at the end of each quarter assuming the following number of days in each quarter? Period Quarter Days in Quarter 1 92 2 92 3 90 4 91 5 92 6 92 7 90 8 91 3

The fixed-rate payment for each quarter is given by: fixed-rate payment = notional amount swap rate number of days in period 360 where the notional amount is $20 million, the swap rate is 7%, and the number of days is the number for that period. For period 1, we have: fixed-rate payment = notional amount swap rate number of days in period 360. Inserting our values, we have: fixed-rate payment for period 1 = 0.07 $20,000,000 92 360 = $357,777.78. Similarly, for periods 2, 3, 4, 5, 6, 7, and 8, the respective fixed-rate payments are: $357,777.78, $350,000.00, $353,888.89, $357,777.78, $357,777.78, $350,000.00, and $353,888.89. (b) Assume that the swap in part (a) requires payments semiannually rather than quarterly. What is the semiannual fixed-rate payment? First, we need the days for each of the four semiannual periods for the two years. Period 1 s days are 92 + 92 = 184. Period 2 s days are: 90 + 91 = 181. Period 3 s days are: 92 + 92 = 184. Period 4 s days are 90 + 91 = 181. We use the formula given above as: fixed-rate payment = notional amount swap rate number of days in period 360. where the notional amount and swap rate are the same but the number of days change as given above. Inserting our values, we get for the first period: 4

fixed-rate payment for period 1 = 0.07 $20,000,000 184 360 = $715,555.56. Similarly, for periods 2, 3, and 4, the respective fixed-rate payments are: $703,888.89, $715,555.56, and $703,888.89. (c) Suppose that the notional amount for the two-year swap is not the same in both years. Suppose instead that in year 1 the notional amount is $20 million, but in year 2 the notional amount is $12 million. What is the fixed-rate payment every six months? The fixed-payments for the first two six-month periods are the same as given in part (b) as $715,555.56 and $703,888.89. For period 3, the fixed-rate payment is: fixed-rate payment for period 3 = 0.07 $12,000,000 184 360 = $429,333.33. Similarly, for period 4, the fixed-rate payment is: fixed-rate payment for period 4 = 0.07 $12,000,000 181 360 = $422,333.33. 16. Given the current three-month LIBOR and the Eurodollar CD futures prices shown in the table below, compute the forward rate and the forward discount factor for each period. Period Days in Quarter 3-month LIBOR Current Eurodollar CD Futures Price 1 90 5.90% 2 91 93.90 3 92 93.70 4 92 93.45 5 90 93.20 6 91 93.15 For period 1, we have: 5

forward rate for period 1 = three-month LIBOR today number of days in period 360. Inserting our values, we get: 5.90% 90 360 = 1.475%. The forward rate for periods 2 through 6 is given by: forward rate = 100 Eurodollar CD Futures price for that period 100 number of days in period. 360 Inserting the value for period 2, we have: forward rate for period 2 = 100 93.90 100 91 360 = 0.015419444 or 1.5419444%. Similarly, for periods 3, 4, 5, and 6, the respective forward rates are: 1.6100%, 1.6738889%, 1.7000%, and 1.7315278%. The forward discount factor for period t is given by: 1 / [(1 + forward rate period 1)(1 + forward rate period 2)... (1 + forward rate period t)]. For period 1, the forward discount factor is: forward discount factor = 1 / (1 + forward rate for period 1) = 1 / 1.01475 = 0.98546440. For period 2, the forward discount factor is: forward discount factor = 1 / [(1 + forward rate period 1)(1 + forward rate period 2)] = 1 / [(1.01475)(1.015419444)] = 0.97049983. Similarly for periods 3, 4, 5, and 6, the respective forward discount factors are: 0.95512236, 0.93939789, 0.92369507, and 0.90797326. 6

17. Answer the following questions. (a) Suppose that at the inception of a five-year interest-rate swap in which the reference rate is three-month LIBOR, the present value of the floating-rate payments is $16,555,000. The fixed-rate payments are assumed to be semiannual. Assume also that the following is computed for the fixed-rate payments (using the notation in the chapter): N notional amount t =1 days in period t forward discount factor for period t = 360 $236,500,000. What is the swap rate for this swap? The swap rate is given by: N t= 1 swap rate = present value of floating-rate payments. days in period t notional amount forward discount factor for period t 360 Inserting our values we get: swap rate = $16,555,000 $236,500,000 = 0.0700 or 7.00%. (b) Suppose that the five-year yield from the on-the-run Treasury yield curve is 6.4%. What is the swap spread? Given the swap rate, the swap spread can be determined. For example, since this is a five-year swap, the convention is to use the five-year on-the-run Treasury rate as the benchmark. Because the yield on that issue is 6.40%, the swap spread is 7.00% 6.40% = 0.6% or 60 basis points. 18. An interest-rate swap had an original maturity of five-years. Today, the swap has two years to maturity. The present value of the fixed-rate payments for the remainder of the term of the swap is $910,000. The present value of the floating-rate payments for the remainder of the swap is $710,000. 7

Answer the following questions. (a) What is the value of this swap from the perspective of the fixed-rate payer? We have: present value of fixed-rate payments = $910,000 and present value of floating-rate payments = $710,000. The two present values are not equal, therefore, for one party the value of the swap increased while for the other party the value of the swap decreased. The fixed-rate payer (or floating-rate receiver) will receive the floating-rate payments. These floating-rate payments have a present value of $710,000. The present value of the payments that must be made by the fixed-rate payer and received by floating-rate payer is $910,000. Thus, the swap has a negative value for the fixed-rate payer equal to the difference in the two present values of $710,000 $910,000 = $200,000.00. This is the value of the swap to the fixed-rate payer. (b) What is the value of this swap from the perspective of the fixed-rate receiver? The floating-rate payer (or fixed-rate receiver) will receive the fixed-rate payments. These fixed-rate payments have a present value of $910,000. The present value of the payments that must be made by the floating-rate payer and received by fixed-rate payer is $710,000. Thus, the swap has a positive value for the floating-rate payer equal to the difference in the two present values of $910,000 $710,000 = $200,000.00. This is the value of the swap to the floating-rate payer. 10. Suppose that a life insurance company has issued a three-year GIC with a fixed-rate of 10%. Under what circumstances might it be feasible for the life insurance company to invest the funds in a floating-rate security and enter into a three-year interest-rate swap in which it pays a floating rate and receives a fixed-rate? If the life insurance can enter a swap that guarantees a satisfactory spread above the 10% it is committed to pay, then it would not only be feasible but desirable to enter into the swap. Suppose the life insurance company can enter into a swap with a bank which has a 8

portfolio consisting of three-year term commercial loans with a fixed interest rate. The principal value of bank s portfolio is $10 million, and the interest rate on all its loans in its portfolio is 11%. The loans are interest-only loans; interest is paid semiannually, and the principal is paid at the end of three years. That is, assuming no default on the loans, the cash flow from the loan portfolio is $550,000 million every six months for the next three years and $10 million at the end of three years (in addition to the $1.1 million interest). To fund its loan portfolio, assume that the bank is relying on the issuance of six-month certificates of deposit. The interest rate that the bank plans to pay on its six-month CDs is six-month LIBOR plus 40 basis points. The risk that the bank faces is that six-month LIBOR will be 10.6% or greater. To understand why, remember that the bank is earning 11% annually on its commercial loan portfolio. If six-month LIBOR is 10.6%, it will have to pay 10.6% plus 40 basis points, or 11%, to depositors for six-month funds and there will be no spread income. Worse, if six-month LIBOR rises above 10.6%, there will be a loss; that is, the cost of funds will exceed the interest rate earned on the loan portfolio. The bank s objective is to lock in a spread over the cost of its funds. The life insurance company can seize this opportunity to cover its commitment to pay a 10% rate for the next three years on the $10 million GIC it has issued. The amount of its GIC is $10 million. Suppose that the life insurance company has the opportunity to invest $10 million in what it considers an attractive three-year floating-rate instrument in a private placement transaction. The interest rate on this instrument is six-month LIBOR plus 160 basis points. The coupon rate is set every six months. The risk that the life insurance company faces in this instance is that six-month LIBOR will fall so that the company will not earn enough to realize a spread over the 10% rate that it has guaranteed to the GIC holders. If six-month LIBOR falls to 8.4% or less, no spread income will be generated. To understand why, suppose that six-month LIBOR at the date the floating-rate instrument resets its coupon is 8.4%. Then the coupon rate for the next six months will be 10% (8.4% plus 160 basis points). Because the life insurance company has agreed to pay 10% on the GIC policy, there will be no spread income. Should six-month LIBOR fall below 8.4%, there will be a loss. We can summarize the asset/liability problems of the bank and the life insurance company as follows. 9

Bank: (i) Has lent long term and borrowed short term. (ii) If six-month LIBOR rises, spread income declines. Life Insurance Company: (i) Has lent short term and borrowed long term. (ii) If six-month LIBOR falls, spread income declines. Now let s suppose the market has available a three-year interest-rate swap with a notional principal amount of $10 million. The swap terms available to the bank are as follows: (i) Every six months the bank will pay 9.45% (annual rate). (ii) Every six months the bank will receive LIBOR. The swap terms available to the insurance company are as follows: (i) Every six months the life insurance company will pay LIBOR. (ii) Every six months the life insurance company will receive 9.40%. What has this interest-rate contract done for the bank and the life insurance company? Consider first the bank. For every six-month period for the life of the swap agreement, the interest-rate spread will be as follows: Annual Interest Rate Received: From commercial loan portfolio 11.00% From interest-rate swap six-month LIBOR Total 11.00% + six-month LIBOR Annual Interest Rate Paid: To CD depositors six-month LIBOR On interest-rate swap 9.45% Total 9.45% + six-month LIBOR Outcome: To be received 11.00% + six-month LIBOR To be paid 9.45% + six-month LIBOR Spread income 1.55% or 155 basis points 10

Thus, whatever happens to six-month LIBOR, the bank locks in a spread of 155 basis points. Now let s look at the effect of the interest-rate swap from the perspective of the life insurance company: Annual Interest Rate Received: From floating-rate instrument 1.6% + six-month LIBOR From interest-rate swap 9.40% Total 11.00% + six-month LIBOR Annual Interest Rate Paid: To GIC policyholders 10.00% On interest-rate swap Six-month LIBOR Total 10.00% + six-month LIBOR Outcome: To be received 11.00% + six-month LIBOR To be paid 10.00% + six-month LIBOR Spread income 1.00% or 100 basis points Thus, whatever happens to six-month LIBOR, the insurance company locks in a spread of 100 basis points. The interest-rate swap has allowed each party to accomplish its asset/liability objective of locking in a spread. It permits the two financial institutions to alter the cash flow characteristics of its assets: from fixed to floating in the case of the bank, and from floating to fixed in the case of the life insurance company. 11