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Swaps. Chapter 7. Goals of Chapter 7. Introduce interest rate (IR) swaps Definition for swaps An illustrative example for IR swaps Discuss reasons for using IR swaps Quotes and valuation of IR swaps Introduce currency swaps

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**Swaps**Chapter 7**Goals of Chapter 7**• Introduce interest rate (IR) swaps • Definition for swaps • An illustrative example for IR swaps • Discuss reasons for using IR swaps • Quotes and valuation of IR swaps • Introduce currency swaps • Payoffs, reasons for using currency swaps, and the valuation of currency swaps • Credit risk of swaps • Other types of swaps**Definition of Swaps**• A swap is an agreement to exchange a series of cash flows (CFs) at specified future time points according to certain specified rules • The first swap contracts were created in the early 1980s • Swaps are traded in OTC markets • Swaps now occupy an important position in OTC derivatives markets • The calculation of CFs depends on the future values of an interest rate, an exchange rate, or other market variables**Interest Rate Swap**• The most common type of swap is a “plain vanilla” IR swap • One party agrees to pay CFs at a predetermined fixed rate on a notional principal for several years • The other party pay CFs at a floating rate on the same notional principal for the same period of time • The floating rate in most IR swaps depends on the LIBORs with different maturities in major currencies • An illustrative example: On Mar. 5 of 2013, Microsoft (MS) agrees to receive 6-month LIBOR and pay a fixed rate of 5% with Intel every 6 months for 3 years on a notional principal of $100 million**Cash Flows of an Interest Rate Swap**---------Millions of Dollars--------- LIBOR Floating Fixed Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, 2013 4.2% Sept. 5, 2013 4.8% +2.10 –2.50 –0.40 Mar.5, 2014 5.3% +2.40 –2.50 –0.10 Sept. 5, 2014 5.5% +2.65 –2.50 +0.15 Mar.5, 2015 5.6% +2.75 –2.50 +0.25 Sept. 5, 2015 5.9% +2.80 –2.50 +0.30 Mar.5, 2016 +2.95 –2.50 +0.45 • For each reference period, the 6-month LIBOR in the beginning of the period determine the payment amount at the end of the period • According to this rule, there is no uncertainty about the first CF exchange • Note that it is not necessary to exchange the principal at any time point • This is why the principal is termed the notional principal (名義本金或名目本金), or just the notional**Cash Flows of an Interest Rate Swap If the Principal was**Exchanged ---------Millions of Dollars--------- LIBOR Floating Fixed Net Date Rate Cash Flow Cash Flow Cash Flow Mar.5, 2013 4.2% Sept. 5, 2013 4.8% +2.10 –2.50 –0.40 Mar.5, 2014 5.3% +2.40 –2.50 –0.10 Sept. 5, 2014 5.5% +2.65 –2.50 +0.15 Mar.5, 2015 5.6% +2.75 –2.50 +0.25 Sept. 5, 2015 5.9% +2.80 –2.50 +0.30 Mar.5, 2016 +102.95 –102.50 +0.45 • If the principal were exchanged at the end of the life of the swap, the nature (or said the net cash flows) of the deal would not be changed in any way • Since only the net CF changes hands in practice for IR swaps, it is not necessary to exchange the principal at any time point • The IR swap can be regarded as the exchange of a fixed-rate bond (with the CFs in the 4th column) for a floating-rate bond (with the CFs in the 3rd column) • This characteristic helps to evaluate the value of an IR swap (introduced later)**Interest Rate Swap**• Day count conventions for IR swaps in the U.S. • Since the LIBOR is a U.S. money market rate, it is quoted on an actual/360 basis • For the fixed rate, it is usually quoted as actual/365 • For the first CF exchange on Slide 7.6, because there are 184 days between Mar. 5, 2013 and Sep. 5, 2013, the accurate CF amounts are (for the floating-rate CF) (for the fixed-rate CF) • For clarity of exposition, this day count issue will be ignored in the rest of this chapter**Interest Rate Swap**• Reasons for using IR swaps • Converting a liability from • fixed rate to floating rate • floating rate to fixed rate • The Intel and Microsoft example 5% LIBOR + 0.1% 5.2% MS Intel LIBOR Original floating-rate debt of MS IR swap Original fixed-rate debt of Intel • The net borrowing rate for Intel’s liability is LIBOR + 0.2% • The net borrowing rate for MS’s liability is 5.1%**Interest Rate Swap**• Converting an asset (or an investment) from • fixed rate to floating rate • floating rate to fixed rate • The Intel and Microsoft example 5% 4.7% LIBOR – 0.2% MS Intel LIBOR Original fixed-rate asset of MS IR swap Original floating-rate asset of Intel • The net interest rate earned for Intel’s asset is 4.8% • The net interest rate earned for MS’s asset is LIBOR – 0.3%**Interest Rate Swap**• When a financial intuition is involved • Usually two nonfinancial companies do not get in touch directly to arrange a swap • It is unlikely for a company to find a trading counterparty which needs the opposite position of the IR swap, i.e., another firm agrees with the principal and maturity but shows a different preference for the floating or fixed IR • In practice, each of them deals with a financial institution (F.I.) 5.015% 4.985% LIBOR + 0.1% 5.2% MS F.I. Intel LIBOR LIBOR Original floating-rate debt of MS Original fixed-rate debt of Intel IR swap IR swap**Interest Rate Swap**5.015% 4.985% 4.7% LIBOR – 0.2% MS F.I. Intel • Note that the F.I. has two separate and offsetting IR swap contracts, and it has to honor the both contracts even Intel or MS defaults • In most cases, Intel and MS do not even know whether the F.I. has entered into an offsetting swap with another firm • In practice, there are many F.I.’s as market markers (or say dealers) in OTC markets and always preparing to trade IR swaps without having an offsetting swap • They need to hedge their unoffset swap positions with Treasury bonds, FRAs, or other IR derivatives LIBOR LIBOR Original fixed-rate asset of MS Original floating-rate asset of Intel IR swap IR swap**Quotes By a Swap Market Maker**• The quotes of IR swaps are expressed as the rate for the fixed-rate side • Bid rate: the fixed rate that the market maker pays for buying (receiving) a series of CFs according to LIBOR • Offer rate: the fixed rate the market marker earns for selling (paying) a series of CFs according to LIBOR • Swap Rate: the fixed rate such that the value of this swap is zero (introduced later), and the bid-offer quotes usually center on the swap rate in practice • The plain vanilla fixed-for-floating swaps are usually structured so that the financial institution earns about 0.03% to 0.04% in the U.S.**Comparative Advantage Argument**Fixed Floating AAA Corp. 4.00% 6-month LIBOR – 0.1% BBB Corp. 5.20% 6-month LIBOR + 0.6% • The comparative advantage argument explains the popularity of the IR swaps • AAA Corp. prefers to borrow at a floating rate and BBB Corp. prefers to borrow at a fixed rate • The fixed or floating IRs they need to pay are • A key feature is that the difference between the two fixed rates (1.2%) is greater than the difference between the two floating rates (0.7%) • AAA (BBB) Corp. has comparative advantage in borrowing fixed-rate (floating-rate) debt**Comparative Advantage Argument**4.35% LIBOR + 0.6% 4% BBB AAA LIBOR Borrow at a floating rate Borrow at a fixed rate IR swap • An ideal win-win solution with a swap • AAA Corp. borrows fixed-rate funds at 4% • BBB Corp. borrows floating-rate funds at LIBOR + 0.6% • Both enter into a fixed-for-floating IR swap to obtain the IRs they prefer • The net borrowing rate for AAA Corp. is LIBOR – 0.35%, which is by 0.25% lower than LIBOR – 0.1% if it borrows at a floating rate directly • The net borrowing rate for BBB Corp. is 4.95%, which is by 0.25% lower than 5.2% if it borrows at a fixed rate directly**Comparative Advantage Argument**4.37% 4.33% LIBOR + 0.6% 4% BBB F.I. AAA LIBOR LIBOR Borrow at a floating rate Borrow at a fixed rate IR swap IR swap • Suppose AAA and BBB cannot deal directly and a F.I. is involved • The net interest rate for AAA Corp. is LIBOR – 0.33%, which is by 0.23% lower than LIBOR – 0.1% if it borrows at a floating rate directly • The net interest rate for BBB Corp. is 4.97%, which is by 0.23% lower than 5.2% if it borrows at a fixed rate directly • The gain of the F.I. is 0.04% • Note that in both cases, the total gains of all participants is 0.5% , which equals (1.2% – 0.7%), where 1.2% (0.7%) is the difference between the fixed (floating) borrowing IRs for AAA and BBB Corp.**Criticism of the Comparative Advantage Argument**• The comparative advantage arises from the unmatched maturities for different IR rates • The 4% and 5.2% rates available to AAA Corp. and BBB Corp. are, for example, 5-year rates • The LIBOR – 0.1% and LIBOR + 0.6% rates are available for only 6 months • The fixed IR level or the spread above or below the LIBOR reflects the creditworthiness of AAA and BBB corporations • Since the 6-month period is so short that the default prob. of BBB is small, BBB can enjoy the comparative advantage on borrowing at a floating rate • In contrast, since lenders intend to cover the default uncertainty for a longer period of time, the 5-year fixed borrowing rate for BBB Corp. is relatively more expensive**Criticism of the Comparative Advantage Argument**• Note that the floating-rate loan will be reviewed (so as the creditworthiness of the borrower) and rolled over every 6 months, so the true cost to borrow at a floating rate depends on the (LIBOR + spread) in the future • If the creditworthiness of BBB deteriorates, the spreads increase or even BBB cannot roll over the loan any more • For the CF schedule on Slide 7.16, the net borrowing rate for BBB Corp. is NOT FIXED at (LIBOR + 0.6%) + 4.37% – LIBOR = 4.97%, but dependent on its future creditworthiness every 6 months • In contrast, if BBB Corp. borrows at a fixed rate, the borrowing rate is fixed at 5.2% for 5 years, regardless of its creditworthiness • As a result, BBB Corp. cannot achieve its goal with a swap perfectly if its creditworthiness changes in the future • The above inference disprove that the comparative advantage argument can explain the popularity of IR swaps**The Nature of Swap Rates**• The n-year swap rate is a constant interest rate corresponding to a credit risk for 2n consecutive 6-month LIBOR loans to AA-rated companies • First, it is known that the 6-month LIBOR is a short-term AA-rating borrowing rate • Second, a F.I. can earn the n-year swap rate by • Lending for the first 6-month loan to a AA borrower and relending it for successive 6-month periods to other AA borrowers for n years, and • Entering into a IR swap to exchange the LIBOR income in the above step for the constant CFs at the n-year swap rate**The Nature of Swap Rates**• Note that the n-year swap rates are lower than n-year AA-rating fixed borrowing rates • For the swap rate, the creditworthiness of the borrowers are always AA for the whole n-year period • For n-year AA-rating fixed borrowing rates, it is only known that the initial creditworthiness of the borrower is AA-rating at the beginning of the n-year period • The lower credit risk for earning the swap rate leads to the lower n-year swap rate than the n-year AA-rating fixed borrowing rate • Together with the fact that the credit risks of AA-rating companies are small in both theory and practice, it can be inferred that the swap rates are close to risk-free**Valuation of IR Swaps**• There are two approaches to price IR swaps • Regard the value of an IR swap as the difference between the values of a fixed-rate bond and a floating-rate bond (see Slide 7.7) • Regard an IR swap as a portfolio of forward rate agreements (FRAs) (For the Intel and MS 3-year IR swap, it can be regarded separately as 6 FRAs)**Valuation of IR Swaps**• Valuation in terms of bond prices • For a swap where fixed CFs are received and floating CFs are paid, its value can be expressed as , where and denote the values of a fixed-rate and floating-rate bonds • The value of a fixed-rate bond (Bfix) can be derived with the traditional discounted cash flow method • The value of a floating-rate bond (Bfl) that pays 6-month LIBOR is always equal to its PAR VALUE immediately after the each payment date**Valuation of IR Swaps**• Note that in any scenario for LIBOR and for different life time of bonds, the Bflis worth its par value on the issue date and on each date immediately after the coupon payment date • Price Bfl(with the face value to be $100) in one scenario for the 6-month LIBOR with 1.5 years to maturity**Valuation of IR Swaps**Value = L = PV of L+k* at t* Value = L Value = L+k* t 0 t* Valuation Date First Pmt Date (Floating Pmt = k*) Second Pmt Date Maturity Date • Generalization for pricing Bfl (with the principal (or said par value) L) • Note that the value of the floating-rate bond at any time point in, for example, the first period is the PV of at , i.e., , where is the continuously compounding zero rate corresponding to the time to maturity of**Valuation of IR Swaps**• An example for pricing IR swaps • For the party to pay the six-month LIBOR and receive fixed 8% (semi-annual compounding) on a principal of $100 million • Remaining life of the IR swap is 1.25 years • LIBOR rates for 3-months, 9-months and 15-months are 10%, 10.5%, and 11% (continuously compounding) • The 6-month LIBOR on the last payment date was 10.2% (semi-annual compounding)**Valuation of IR Swaps**• For per $100 principal • The coupon payment of after 3 months is • The value of today is according to the formula on Slide 7.23**Valuation of IR Swaps**• Valuation in terms of FRAs • Each exchange in an IR swap is an FRA • Note that for a newly issued IR swap, the first exchange of payments is known when the swap is negotiated • For each of other exchanges, it can be regarded as a FRA applied for a future period of 6 months • Recall that for a FRA applied in , the payoff of the lender at is (see Slide 4.35), where is the principal, is the fixed IR specified in the FRA contract, and is the actual LIBOR in • Considering a pay-floating-receive-fixed IR swap with the principal , for each 6 months, the swap holder can receive the net payoff of , where is the actual 6-month LIBOR for that period and is the fixed IR specified in the swap contract**Valuation of IR Swaps**• The above formula can derive the value of each exchange in an IR swap • Note that the value of any derivatives equals the present value of it expected payoff • This feature has been used to price FRA on Slide 4.38 • To evaluate the expected payoff of a FRA, the expectation of the future LIBOR is needed Value of FRA = = = • It is known that the expected future LIBORs equal the forward rates ) based on today’s term structure of IRs: Value of FRA = =**Valuation of IR Swaps**(cont. comp.) 11.044% (semi-annual comp.) Expected cash outflow at is (cont. comp.) 12.102% (semi-annual comp.) Expected cash outflow at is • Consider the pay-floating-receive-fixed IR swap example on Slide 7.24. For per $100 principal,**Valuation of IR Swaps**• An IR swap is worth zero when it is first initiated • When a swap contract is first negotiated, the swap rate is determined such that the value of the swap is zero initially • This feature is similar to set the delivery prices of futures contracts to be the futures prices such that the futures contracts are worth zero when they are initiated • Due to, a swap rate equals a par yield (see the example on Slide 7.33) • With the passage of time, the value of an IR swap emerges and can be either positive or negative • One party’s gains are the other party’s losses, so two parties of a swap have opposite points of view on the swap value**Valuation of IR Swaps**• Although the swap is zero initially, it does not mean that the value of each individual FRA is zero initially • The initial zero value of a swap actually means that the sum of the values of all FRAs underlying the swap is zero • For a pay-fixed-receive-floating swap on the issue date, • If the zero curve is upward sloping forward rates ↑ with T • The forward rates with shorter time to maturities < the swap rate negative values for FRAs with shorter time to maturities • The forward rates with longer time to maturities > the swap rate positive values for FRAs with longer time to maturities • If the zero curve is downward sloping forward rates ↓with T • The forward rates with shorter time to maturities > the swap rate positive values for FRAs with shorter time to maturities • The forward rates with longer time to maturities < the swap rate negative values for FRAs with longer time to maturities**Determine LIBOR Zero Curve with Eurodollar Futures and Swaps**• Construct the LIBOR zero curve (Note that derivatives traders commonly use LIBOR as proxies for risk-free rates when trading derivatives) • : the quotes of spot LIBOR provided by financial institutions are used • in (sometimes ): the quotes of Eurodollar futures are used to derive the LIBOR zero rates • Suppose the zero rate for is known • With the convexity adjustment, the forward rates () for can be derived from the futures rates implied from the quotes of Eurodollars futures • Finally, we can deduce through**Determine LIBOR Zero Curve with Eurodollar Futures and Swaps**• For longer : the quotes of swap rates are used to derive the LIBOR zero rate • Consider a pay-floating-receive-fixed IR swap with the swap rate of 5%, principal of $100, and 2 years to maturity • Suppose the 6-month, 12-month, and 18-month LIBOR zero rates are 4%, 4.5%, and 4.8% with cont. compounding • Since the initial value of a swap is zero, then and thus • Solve for the 2-year zero rate to be 4.953% • Similar to the bootstrap method, the zero LIBOR rates for shorter should be solved first before solving the zero LIBOR rates for longer**Overnight Indexed Swaps (OISs)**• An OIS is a swap where a fixed rate for a period is exchanged for the geometric average of the overnight rates during the period • The fixed rate is referred to as the OIS rate • OISs tend to have short lives ( 3 months) • Longer-term OISs are typically divided into three-month sub-periods • At the end of each sub-period, the net of the actual geometric average of the overnight rates during the sub-period and the fixed OIS rate will be exchange • Should the OIS rate equal the LIBOR rate? Consider the following trading strategy for a AA-rated bank:**Overnight Indexed Swaps (OISs)**• Borrow $100 in the overnight market for 3 months (92 days for example), rolling the interest and principal on the loan forward each night (pay the geometric average of the overnight rates) • Enter into an OIS to convert the geometric average of the overnight rates to the 3-month OIS rate • Lend the borrowed $100 to another AA-rated financial institution for three months at LIBOR ※ Final payoff = $100 (92/365) (LIBOR OIS rate) • In fact, the OIS rate is lower than the LIBOR • This is because OIS rate is a continually refreshed overnight rate (always lend or borrow with AA financial institutions) • To earn 3-month LIBORs, the bank should bear the default risk of its trading counterparty, which is AA-rated initially • The OIS rate is even closer to the risk-free interest rate**Overnight Indexed Swaps (OISs)**• In practice, many derivatives dealers choose to use OIS rates for discounting collateralized transactions (less risky) and use LIBORs for discounting noncollateralized transactions (more risky) • The (LIBOR – OIS) spread • Defined as the 3-month LIBOR rate over the 3-month OIS rate • Can be used to measure the degree of stress in financial markets • In normal market condition, this spread is about 10 basis points • In Oct. 2009, this spread spiked to an all-time high of 364 basis points because banks are reluctant to lend to each other for three-month periods • In Dec. 2011, due to the concern of the crisis in Greece, this spread rose to 50 basis points**Determine Zero Curve Using OISs**• Similar to the method for constructing the LIBOR zero curve, we can derive zero curve using OIS quotes • months: the quotes of OIS rates provided by financial institutions are used • For a longer (when there are periodic settlements (usually every 3 months) in OIS contracts) • The OIS rate approximately defines a par yield bond • For a 1.25-year OIS contract with the OIS rate to be 4%, it can be regarded as a bond paying a quarterly coupon at a rate of 4% per annum and sold at par • Suppose the 3-, 6-, 9-, and 12-month OIS zero rates are 3%, 3.5%, 4%, and 4.5% with continuous compounding**Determine Zero Curve Using OISs**• This OIS contracts implies that • Solve for the 1.25-year OIS zero rate to be 3.9798% • For a is so long such that the quotes of OIS rates are not available or unreliable, e.g., years • Note that LIBOR IR swaps are traded for longer maturities than OIS • Assume the (LIBOR – OIS) spread is constant and as it is for the longest OIS maturity for which there is reliable data, e.g., the 5-year OIS contract and the corresponding (LIBOR – OIS) spread is 20 basis points • Use the LIBOR zero curve minus the constant (LIBOR – OIS) spread to derive the OIS rate zero curve**Currency Swap**• Currency swap is another popular type of swaps • It involves exchanging principal and interest payments in one currency for principal and interest payments in another currency • Different from IR swaps, the principal amounts (in different currencies) are exchanged at the beginning and at the end of the life of a currency swap • The principal amounts are chosen to be approximately equivalent using the exchange rate at the swap’s initiation • An example of a currency swap: IBM pays 5% on a principal of £10,000,000 and receive 6% on a principal of $15,000,000 from British Petroleum (BP) every year for 5 years**Currency Swap**£10 mil. BP IBM $15 mil. Dollar 6% BP IBM Sterling 5% ※ A currency swap can be regarded as two concurrent loans denominated in different currencies ※ Since the values of $15 mil. and £10 mil. are equivalent initially, the net value of these two concurrent loans are zero at the swap’s initiation $15 mil. BP IBM £10 mil.**Currency Swap**• Typical uses of a currency swap is to • Convert a liability in one currency to a liability in another currency • Convert an investment in one currency to an investment in another currency Dollar 6% Sterling 5% Dollar 6% BP IBM Sterling 5% Dollar 6% Dollar 6% BP IBM Sterling 5% Sterling 5%**Comparative Advantage Arguments for Currency Swaps**• The comparative advantage argument explains the popularity of the currency swaps • General Electric (GE) prefers to borrow AUD and Qantas Airways (QA) prefers to borrow USD • The USD and AUD borrowing IRs they face are • GE has a comparative advantage in the USD market, whereas QA has a comparative advantage in the AUD market USD AUD General Electric 5.0% 7.6% Qantas Airways 7.0% 8.0%**Comparative Advantage Arguments for Currency Swaps**• Exploit the comparative advantage with currency swaps • Suppose that GE intends to borrow 20 mil. AUD and QA intends to borrow 15 mil. USD, and the current exchange rate is 0.75USD per AUD • GE borrows USD, QA borrows AUD, and they use currency swaps to transform GE’s USD loan into a AUD loan and QA’s AUD loan into a USD loan • GE pays 6.9% in AUD (0.7% better off) and QA pays 6.3% in USD (0.7% better off) USD 6.3% USD 5.0% AUD 8.0% USD 5.0% QA F.I. GE AUD 8.0% AUD 6.9%**Comparative Advantage Arguments for Currency Swaps**• Different ways to arrange the currency swaps • QA bears some foreign exchange risk • GE bears some foreign exchange risk • These two alternatives are unlikely to be used in practice because the firms prefer to eliminate the foreign exchange risk with currency swaps thoroughly USD 6.3% USD 5.2% USD 5.0% USD 6.1% AUD 8.0% AUD 8.0% USD 5.0% USD 5.0% QA QA F.I. F.I. GE GE AUD 8.0% AUD 6.9% AUD 8.0% AUD 6.9%**Valuationof Currency Swaps**• Like IR swaps, currency swaps can be valued either as the difference between 2 bonds or as a portfolio of forward contracts • Valuation in terms of bond prices • For a receive-dollar-pay-foreign-currency currency swap, then , where is the domestic bond defined by the remaining USD CFs, is the bond defined by the remaining foreign-currency CFs, and is the spot exchange rate (expressed as dollars for per unit of foreign currency) • In contrast, for a pay-dollar-receive-foreign-currency currency swap, then**Valuationof Currency Swaps**• An example for pricing currency swaps • All Japanese LIBOR zero rates are 4% (foreign IR) • All USD LIBOR zero rates are 9% (domestic IR) • A currency swap is to received 5% in yen and pay 8% in dollars. Payments are made annually • Principals are $10 million and 1,200 million yen • Swap will last for 3 more years • Current exchange rate is 110 yen per dollar**Valuationof Currency Swaps**• (million $)**Valuationof Currency Swaps**• Valuation in terms of forward contracts • Each exchange of payments in a fixed-for-fixed currency swap is a foreign exchange (FX) forward contract • FX forwards is an agreement to trade an amount of a foreign currency at the specified price on a predetermined future date • FX forwards are similar to the foreign currency futures contracts (introduced in Ch. 5) except that FX forwards are traded in OTC markets and thus there is no daily settlement requirement • Suppose the principal is and the specified trading price is (expressed as domestic dollars / per foreign dollar) • Payoff of a FX forwards to purchase foreign currency at is , where is the domestic-dollar price of the foreign currency (or the FX rate) at • The value of a FX forward is the PV of its expected payoff**Valuationof Currency Swaps**• The forward (or futures) price of the foreign currency provide the unbiased approximation for based on the information of IRs today • Since the forward price of the foreign currency is (introduced on Slides 5.23-5.24), we can obtain (In other words, the expected future FX rate equals the forward FX rate today) • The forward FX rates of Japanese Yen in the example on Slide 7.41 are

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