Finance Case Study

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Chapter7-Swaps.pdf

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2/26/20

Chapter 7

Swaps

• A swap is an over-the-counter derivatives agreement between two companies to exchange cash flows in the future.

• The agreement defines the dates when the cash flows are to be paid and the way in which they are to be calculated.

• Usually the calculation of the cash flows involves the future value of an interest rate, an exchange rate, or other market variable.

• A forward contract can be viewed as a simple swap. • Whereas a forward contract is equivalent to the exchange of cash flows on just on future

date, swaps typically lead to cash-flow exchanges taking place on several future dates.

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Swaps - Fundamentals • So the basic idea of a swap:

• agree to exchange interest payments of different kinds • different interest computations • different currencies

• during a given time period (settlement period) on certain dates (settlement dates) • with interest payments computed on notional amount • with a predetermined termination date

• Notional amount • never exchanges hands in single-currency interest rate swaps à parties agree to exchange

only the net amount (netting)

• Reference interest rate • LIBOR = reference floating rate in most cases

Mechanism of interest rate swaps

• By far the most common over-the-counter derivative is a ‘‘plain vanilla’’ interest rate swap.

• In this a company agrees to pay cash flows equal to interest at a predetermined fixed rate on a notional principal for a number of years.

• In return, it receives interest at a floating rate on the same notional principal for the same period of time.

• LIBOR • The floating rate in most interest rate swap agreements is the London Interbank Offered

Rate (LIBOR)

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Swap structure • Without intermediary

• With Intermediary

Firm A Firm B

Firm A

Bank

Firm B

Example

• Consider a hypothetical three-year swap initiated on March 8, 2016, between Apple and Citigroup. We suppose Apple agrees to pay to Citigroup an interest rate of 3% per annum on a notional principal of $100 million, and in return Citigroup agrees to pay Apple the six-month LIBOR rate on the same notional principal.

• Apple is the fixed-rate payer ; Citigroup is the floating-rate payer . • Assume the agreement specifies that payments are to be exchanged every six

months and that the 3% interest rate is quoted with semiannual compounding.

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Example

• The first exchange would take place on September 8, 2016, six months after the initiation of the agreement.

• Citigroup would pay Apple interest on the $100 million principal at the six- month LIBOR rate prevailing six months prior to September 8, 2016—that is, on March 8, 2016. Suppose that the six-month LIBOR rate on March 8, 2016, is 2.2% ($1.1 million)

• Apple would pay Citigroup $1.5 million. This is the interest on the $100 million principal for six months at a rate of 3% per year ($1.5 million)

• Note that there is no uncertainty about this first exchange of payments because it is determined by the LIBOR rate at the time the contract is agreed to.

Example

• The second exchange of payments would take place on March 8, 2017, one year after the initiation of the agreement. Apple would pay $1.5 million to Citigroup.

• Suppose that the six-month LIBOR rate on September 8, 2016, proves to be 2.8%. Citigroup pays $1.4 million to Apple.

• Suppose there are 6 exchanges:

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Interest rate swaps

• Plain-vanilla Interest Rate Swap • Contract by which

• Buyer (long) is committed to pay fixed rate R (similar to FRA: buyer locks in borrowing rate)

• Seller (short) is committed to pay variable r (e.g., LIBOR)

• on notional • no exchange of principal

• at future dates set in advance • t + Dt, t + 2 Dt, t + 3Dt , t+ 4 Dt, ...

• most common swap : 6-month LIBOR (Dt=6 months)

Why swaps? Evolution of Swaps—A brief History

• Increase in exchange rate volatility (1972) • increase in earnings volatility • fluctuation in asset value due to exchange rate volatility

• The Solution--parallel loans • two firms simultaneously make financial loans to each other • increasing use in the 1970’s

• but difficult to find partners • ~1981 swaps written by banks to help firms conduct parallel loan transactions

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Economic Benefits of Swaps 1. Financial Arbitrage

• Differential currency borrowing rates • Differing fixed/floating borrowing rates

2. Tax and Regulatory Arbitrage 3. Managing interest rate or currency risk

• May be cheaper than alternatives (futures, for instance)

4. Completing markets • No available alternatives • e.g. originally no interest rate futures, just swaps

Using the Swap to Transform a Liability

• As shown the previous example, for Apple, the swap could be used to transform a floating-rate loan into a fixed-rate loan.

• Suppose that Apple has arranged to borrow $100 million for three years at LIBOR plus 10 basis points ( LIBOR plus 0.1%.)

• After Apple has entered into the swap with the following three sets of cash flows: 1. It pays LIBOR plus 0.1% to its outside lenders. 2. It receives LIBOR under the terms of the swap. 3. It pays 3% under the terms of the swap.

• These three sets of cash flows net out to an interest rate payment of 3.1% • Thus, for Apple the swap could have the effect of transforming borrowings at a

floating rate of LIBOR plus 10 basis points into borrowings at a fixed rate of 3.1%.

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Using the Swap to Transform a Liability

• A company wishing to transform a fixed-rate loan into a floating-rate loan would enter into the opposite swap.

• Suppose that Intel has borrowed $100 million at 3.2% for three years and wishes to switch to a floating rate linked to LIBOR.

• Like Apple it contacts Citigroup. We assume that it agrees to enter into the following swap to convert to a floating rate:

1. It pays 3.2% to its outside lenders. 2. It pays LIBOR under the terms of the swap. 3. It receives 2.97% under the terms of the swap.

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Using the Swap to Transform an Asset

• Swaps can also be used to transform the nature of an asset. Consider Apple in our example. • The swap in could have the effect of transforming an asset earning a fixed rate of interest

into an asset earning a floating rate of interest. • Suppose that Apple owns $100 million in bonds that will provide interest at 2.7% per annum

over the next three years. • After Apple has entered into the swap, it is in the position .It has three sets of cash flows:

1. It receives 2.7% on the bonds. 2. It receives LIBOR under the terms of the swap. 3. It pays 3% under the terms of the swap.

• These three sets of cash flows net out to an interest rate inflow of LIBOR minus 30 basis points.

Changing the nature of an asset – one more example • Suppose that Microsoft owns $100

million in bonds that will provide interest at 4.7% per annum over the next 3 years. Microsoft could use a swap to transform an asset earning 4.7% into an asset earning LIBOR minus 30 basis points.

• Microsoft can enter into a swap, it with the following three sets of cash flows:

1. It receives 4.7% on the bonds. 2. It receives LIBOR under the terms of the swap. 3. It pays 5% under the terms of the swap.

• Suppose that Intel has an investment of $100 million that yields LIBOR minus 20 basis points. Intel wants to transform an asset earning LIBOR minus 20 basis points into an asset earning 4.8%.

• Intel can enter into a swap, it with the following three sets of cash flows:

1. It receives LIBOR minus 20 basis points on its investment. 2. It pays LIBOR under the terms of the swap. 3. It receives 5% under the terms of the swap.

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Role of Financial Intermediary

• Firm usually do not enter into swaps with each other but use intermediaries. • A market maker such as Citigroup provides the full set of quotes for plain

vanilla U.S. dollar swaps. • ‘‘Plain vanilla’’ LIBOR-for-fixed swaps on US interest rates are usually

structured so that the financial institution earns about 3 or 4 basis points (0.03% or 0.04%) on a pair of off setting transactions.

Organization of Trading

• A market maker such as Citigroup provides the full set of quotes for plain vanilla U.S. dollar swaps.

• A Swap Rate is the average of bid and offer

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Day count issues (briefly cover) • The day count conventions affect payments on a swap, and some of the numbers calculated in the

examples we have given do not exactly reflect these day count conventions.

• In general, a LIBOR-based floating-rate cash flow on a swap payment date is calculated as LRn/360, where L is the principal, R is the relevant LIBOR rate, and n is the number of days since the last payment date.

• The fixed rate that is paid in a swap transaction is similarly quoted with a particular day count basis being specified. As a result, the fixed payments may not be exactly equal on each payment date.

• The fixed rate is usually quoted as actual/365 or 30/360. It is not therefore directly comparable with LIBOR because it applies to a full year.

• To make the rates approximately comparable, either the 6-month LIBOR rate must be multiplied by 365/360 or the fixed rate must be multiplied by 360/365.

The comparative advantage argument

• An explanation commonly put forward to explain the popularity of swaps concerns comparative advantage.

• Some companies, it is argued, have a comparative advantage when borrowing in fixed-rate markets, whereas other companies have a comparative advantage when borrowing in floating-rate markets.

• To obtain a new loan, it makes sense for a company to go to the market where it has a comparative advantage.

• As a result, the company may borrow fixed when it wants floating, or borrow floating when it wants fixed.

• The swap is used to transform a fixed-rate loan into a floating-rate loan, and vice versa.

Difference between the two fixed rates and floating rates are not the same

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Critique

• Should the spreads between the rates offered to AAACorp and BBBCorp be different in fixed and floating markets?

• Now that the interest rate swap market has been in existence for a long time, we might reasonably expect these types of differences to have been arbitraged away.

• The reason that spread differentials appear to exist is due to the nature of the contracts available to companies in fixed and floating markets

• In the floating-rate market, the lender usually has the opportunity to review the floating rates every 6 months à if the creditworthiness of AAACorp or BBBCorp has declined, the lender has the option of increasing the spread over LIBOR that is charged.

• The providers of fixed-rate financing do not have the option to change the terms of the loan in this way

ST vs LT

• The spreads between the rates offered to AAACorp and BBBCorp are a reflection of the extent to which BBBCorp is more likely than AAACorp to default.

• During the next 6 months, there is very little chance that either AAACorp or BBBCorp will default.

• As we look further ahead, the probability of a default by a company with a relatively low credit rating (such as BBBCorp) is liable to increase faster than the probability of a default by a company with a relatively high credit rating (such as AAACorp).

• This is why the spread between the 5-year rates is greater than the spread between the 6-month rates.

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The nature of swap rates

• Let’s examine the nature of swap rates and the relationship between swap and LIBOR markets.

• LIBOR is the rate of interest at which AA-rated banks borrow for periods up to 12 months from other banks.

• A swap rate is the average of • (a) the fixed rate that a swap market maker is prepared to pay in exchange for receiving

LIBOR (its bid rate) and • (b) the fixed rate that it is prepared to receive in return for paying LIBOR (its offer rate).

• Like LIBOR rates, swap rates are not risk-free lending rates. However, they are reasonably close to risk-free in normal market conditions

The nature of swap rates

• A financial institution can earn the 5-year swap rate on a certain principal by doing the following:

1. Lend the principal for the first 6 months to an AA borrower and then relend it for successive 6-month periods to other AA borrowers; and

2. Enter into a swap to exchange the LIBOR income for the 5-year swap rate

• This shows that the 5-year swap rate is an interest rate with a credit risk corresponding to the situation where 10 consecutive 6-month LIBOR loans to AA companies are made.

• Note that 5-year swap rates are less than 5-year AA borrowing rates. • It is much more attractive to lend money for successive 6-month periods to borrowers who

are always AA at the beginning of the periods than to lend it to one borrower for the whole 5 years when all we can be sure of is that the borrower is AA at the beginning of the 5 years à swap rates are continuously “refreshed” LIBOR rates.

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Valuation of interest rate swaps

• An interest rate swap is worth close to zero when it is first initiated. After it has been in existence for some time, its value may be positive or negative.

• Each exchange of payments in an interest rate swap is a forward rate agreement (FRA) where interest at a predetermined fixed rate is exchanged for interest at the LIBOR floating rate.

• FRA can be valued by assuming that forward rates are realized. Because it is nothing more than a portfolio of FRAs, an interest rate swap can also be valued by assuming that forward rates are realized. The procedure is:

1. Calculate forward rates for each of the LIBOR rates that will determine swap cash flows. 2. Calculate the swap cash flows on the assumption that LIBOR rates will equal forward rates. 3. Discount these swap cash flows (using the LIBOR/swap zero curve) to obtain the swap value.

Example – valuing interest rate swap • Suppose corporations A and B enter into the following swap agreement:

• Notional amount: $10,000,000 • A makes semiannual payments to B of 3% of the notional • B makes semiannual payment to A of LIBOR+0.5% (this is a 6-month LIBOR). • This contract is equivalent to 4 forward contracts

• Suppose that the forward rates are: – Current LIBOR rate = 2.0% – 6 to 12 month rate = 2.5% – 12 to 18 month rate = 3.0% – 18 to 24 month rate = 3.5% – 24 to 30 month rate = 4.0%

Pay- ments

6 months 1 year 18 months 2 years

A to B $0.3m $0.3m $0.3m $0.3m

B to A $10m at LIBOR0+0.5%

$10m at LIBOR6+0.5%

$10m at LIBOR12+0.5%

$10m at LIBOR18+0.5%

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Interest Rate Swap Pricing

• Sum all discounted cash flows A to B (similarly to bond valuation we use the corresponding ”spot” rate for each future period – here forward LIBOR rate):

• CF1 = -10m*0.03/(1.025) = -0.2927m • CF2 = -10m*0.03/[(1.025)(1.03)] = -0.2842m • CF3 = -10m*0.03/[(1.025)(1.03)(1.035)] = -0.2745m • CF4 = -10m*0.03/[(1.025)(1.03)(1.035)(1.04)] = -0.2640m

Sum -1.1154m

• Similarly, B’s floating rate payments to A • CF1 = -10m*0.025/(1.025) = 0.2439 • CF2 = -10m*0.030/[(1.025)(1.03)] = 0.2842 • CF3 = -10m*0.035/[(1.025)(1.03)(1.035)] = 0.3203 • CF4 = -10m*0.040/[(1.025)(1.03)(1.035)(1.04)] = 0.3520

Sum -1.2004

Interest Rate Swap Pricing

• Value to A is net flow to A: $1.2004m - $1.1154m = $0.085m

• What happens if LIBOR rates increase more than expected? • B’s payments increase • Good for A, Bad for B

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How the value changes through time

• The fixed rate in an interest rate swap is chosen so that the swap is worth zero initially. This means that the sum of the values of the FRAs underlying the swap is initially zero.

• It does not mean that the value of each individual FRA is zero. In general, some FRAs will have positive values while others will have negative values.

• Recall,

Fixed-for-fixed Currency Swaps

• Involves exchanging principal and interest payments at a fixed rate in one currency for principal and interest payments at a fixed rate in another currency.

• A currency swap agreement requires the principal to be specified in each of the two currencies.

• The principal amounts in each currency are usually exchanged at the beginning and at the end of the life of the swap.

• Usually the principal amounts are chosen to be approximately equivalent using the exchange rate at the swap’s initiation.

• But when they are exchanged at the end of the life of the swap, their values may be quite different.

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Example – flat term structure and continuous compounding

• The term structure of risk-free interest rates is flat in both Japan and the United States. The Japanese rate is 1.5% per annum and the U.S. rate is 2.5% per annum (both with continuous compounding).

• A financial institution has entered into a currency swap in which it receives 3% per annum in yen and pays 4% 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 current exchange rate is 110 yen = $1.

Continuous discounting at 2.5% and 1.5%

Example – flat term structure and continuous compounding

• Vswap = S0 x BF - BD • value of a swap where the foreign currency is received, and dollars are paid

• where BF is the value, measured in the foreign currency, of the bond defined by the foreign cash flows on the swap and BD is the value of the bond defined by the domestic cash flows on the swap, and S0 is the spot exchange rate (expressed as number of dollars per unit of foreign currency)

• The value of the swap in dollars is therefore (1252/110)-10.491= 0.9629 million

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Currency Swaps – more rigorous approach

• Company A • Wants to borrow ¥1 billion

• Company J • Wants to borrow $10 million

• Same time frame (2 years) • Assume (for simplicity) that ¥1 = $0.01

• Swap contract • Semi-annual payments (USD forward rates: 2.5%, 3%, 3.5% & 4%) • fixed-for-fixed swap

• A pays 2.5% on yen • J pays 3.0% on dollars

Currency Swap Cash Flows

Swap Value 1. Forward prices

– This contract is equivalent to 4 forward contracts 2. Transaction costs

– Bank gets paid, lessens the value of a swap 3. Default risk

– Counterparty risk reduces swap value

Payments 6 months 1 year 18 months 2 years

A to J ¥25m ¥25m ¥25m ¥1025m

J to A $0.3m $0.3m $0.3m $10.3m

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Currency Swap Value

• Vswap = BD – S0 x BF • where BF is the value, measured in the foreign currency, of the bond defined by

the foreign cash flows on the swap and BD is the value of the bond defined by the domestic cash flows on the swap, and S0 is the spot exchange rate (expressed as number of dollars per unit of foreign currency)

• Value of A payments to J

• Sum of discounted cash flows

• Assume forward prices for yen are • $0.0105 • $0.0110 • $0.0115 • $0.0120

• Here we use forward price equivalents for cash flows to discount

Currency Swap Pricing

• Sum all discounted cash flows (in $): • CF1 = -25*0.0105/(1.025) = -0.2561 • CF2 = -25*0.0110/[(1.025)(1.03)] = -0.2605 • CF3 = -25*0.0115/[(1.025)(1.03)(1.035)] = -0.2631 • CF4 = -1025*0.0120/[(1.025)(1.03)(1.035)(1.04)] = -10.8236

Sum -11.6033

• Similarly for J’s payments to A (in $): • CF1 = 0.3/(1.025) = 0.2927 • CF2 = 0.3/[(1.025)(1.03)] = 0.2842 • CF3 = 0.3/[(1.025)(1.03)(1.035)] = 0.2745 • CF4 = 10.3/[(1.025)(1.03)(1.035)(1.04)] = 9.0636

Sum 9.9150

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Currency Swap Pricing

• All CFs from Company A’s perspective, so the value of swap for Company A is

$9.915m – $11.6033m = -$1.6883m

• Require payment up front for entering the swap! • Value of yen expected to rise relative to dollar over the next 2

years • Note:

• Some examples in the book use net cash flows--combines all CFs in each period after converting to $ equivalents

Currency Swaps (cont.)

• If yen rises more than expected, what happens? • A has locked in forward rates • Gets payment up front • Rising yen makes $ less valuable

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Credit/Default Risk

• The credit risk arises from the possibility of a default by the counterparty when the value of the contract to the financial institution is positive.

• Not the same thing as market risk • arises from the possibility that market variables such as interest rates and

exchange rates will move in such a way that the value of a contract to the financial institution becomes negative.

• Market risks can be hedged by entering into off setting contracts; credit risks are less easy to hedge.

• Can use a CDS - like an insurance contract that pays off if a particular company or country defaults

• Priced into the discount rates above