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Faculty of Business and Economics

University of Hong Kong

Dr. Huiyan Qiu

MFIN6003 Derivative Securities

Lecture Note Six

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OutlineIntroduction to swaps using an example of a commodity swap.

• Swap settlement; swap counterparty; market value of a swap; computing swap price (rate).

Interest Rate Swaps

Currency Swaps

Appendix: more on swaps

• Swaptions

• Total Return Swaps

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Introduction to Swaps

A swap is a contract calling for an exchange of payments over time

• The swap payments are determined by the difference in swap price and market price over time

• A swap provides a means to hedge a stream of risky payments (lower transaction cost)

• A single-payment swap is the same thing as a cash-settled forward contract

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An Example of a Commodity Swap

An industrial producer, IP Inc., needs to buy 100,000 barrels of oil 1 year from today and 2 years from today (concern? P↑)

How to hedge against the risk in oil cost?

Relevant information:

• The forward prices for deliver in 1 year and 2 years are $20 and $21/barrel.

• The risk-free 1- and 2-year zero-coupon bond yields are 6% and 6.5%

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A Commodity Swap (cont’d)Strategy 1: Long forward contracts for 100,000 barrels in each of the next 2 years

• IP pays $20 in year one and $21 in year two for oil

Strategy 2: Prepaid swap

• A single payment today for multiple deliveries of oil in the future.

• IP pays an oil supplier $37.383 per barrel in exchange for a commitment to delivering one barrel in each of the next two years. Credit risk?

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A Commodity Swap (cont’d)

Strategy 3: Swap

• Defer payments until the oil is delivered, while still fixing the total price

• A swap usually calls for equal payment in each year

• The 2-year swap price is $20.483

Any series of payments that have a PV of $37.383 is acceptable (ignoring the credit risk)

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Time Line – Payments

0 21

Unhedged S1 S2

Strategy 1 20 21

Strategy 2 37.873

Strategy 3 20.483 20.483

All series of payments have a PV of $37.383.

Swaps, Forwards, and Financing

Swaps are nothing more than forward contracts coupled with borrowing and lending money

• Compare the swap price and the forward prices, we are overpaying by $0.483 in the first year, and we are underpaying by $0.517 in the second year

• We are lending the counterparty money for 1 year. The interest rate on this loan is

0.517/0.483 – 1 = 7%.

• Given 1- and 2-year zero-coupon bond yields of 6% and 6.5%, 7% is the 1-year implied forward yield from year 1 to year 2. (Fair pricing!)

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No-Arbitrage Principle

The present value of future payments for the same series of future commodity delivery should be the same. Otherwise, arbitrage!

If the present value of payment using two forward contracts for the oil delivery in year 1 and in year 2 is lower than that of payment using two-year swap, one can long the two forward contracts and short the swap to gain the arbitrage profit.

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Computing the Swap PriceSuppose there are n swap settlements, occurring on dates ti, i = 1,… , n. What is the swap price R?

PV (swap price) = PV (forward price)

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Physical vs. Financial Settlement

The results for the buyer are the same whether the swap is settled physically or financially. In both cases, the net cost of the buyer is fixed at the swap price of $20.483, whatever the market price of oil.

Physical settlement

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Financial SettlementThe oil buyer, IP, pays the swap counterparty the difference between $20.483 and the spot price, and the oil buyer then buys oil at the spot price

100,000 barrels are the notional amount of the swap, used to determine the magnitude of the payments when the swap is settled financially

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The Swap Counterparty

The swap counterparty is a dealer, who is, in effect, a broker between buyer and seller

The fixed price paid by the buyer, usually, exceeds the fixed price received by the seller. This price difference is a bid-ask spread, and is the dealer’s fee

Back-to-back transaction or “matched book” transaction: the situation where the dealer matches the buyer and seller.

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Matched Book Transaction

The dealer bears the credit risk of both parties, but is not exposed to price risk

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Unmatched Book Transaction

In the case that the swap transaction is not matched, the dealer serves as counterparty to the oil buyer and is facing future oil price risk: obligation to receive fixed price and pay floating price.

Swap Counterparty

Spot price - $20.483

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Unmatched Book TransactionTo hedge the swap transaction with the buyer, the dealer can enter into long forward or futures contracts.

Cash flows:

The net cash flow for the hedged dealer is a loan

Thus, the dealer also has interest rate exposure (which can be hedged by using Eurodollar futures or FRAs)

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The Market Value of a Swap

The market value of a swap is zero at interception

• Once the swap is struck, its market value will generally no longer be zero (change in forward price, in interest rate…)

• Even if there is no change in interest rates or the forward prices, the swap changes value after payment.

The market value of the swap is the difference in the PV of payments between the original and new swap prices

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The Market Value of a Swap

Example: change in forward price

• Assume immediately after the initiation of the swap, the forward curve for oil rises by $2 in both years

• Assume interest rates are unchanged

• The new swap price will be $22.483, $2 higher than the old one (check and understand why exactly $2)

• PV of the differences = 2/1.06 + 2/(1.0652) = $3.65

• $3.65 is the market value of the old swap

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Misuse of Swap: Enron’s Case

Energy giant Enron collapsed in 2001.

As charged by SEC, other companies helped Enron mislead investors.

One case: J.P Morgan Chase had helped Enron characterize loan proceeds as operating income by using swaps.

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Enron’s Hidden Debt

Figure Enron’s swaps with Mahonia and Chase. Source: Securities and Exchange Commission.

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Kinds of SwapsInterest Rate Swaps: payments are the difference of interest payments based on floating rate and fixed rate (swap rate)• The notional principle of the swap is the amount

on which the interest payments are based

• The life of the swap is the swap term or swap tenor

Currency Swaps: entail an exchange of payments in different currencies• A currency swap is equivalent to borrowing in

one currency and lending in another

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An Example of an Interest Rate Swap

XYZ Corp. has $200M of floating-rate debt at LIBOR, i.e., every year it pays that year’s LIBOR. XYZ would prefer to have fixed-rate debt with 3 years to maturity

• Retire the floating-rate and issue fixed rate debt (Transaction cost? Feasible? )

• Enter a strip of FRAs (FRA rates for each year varies)

• Enter a swap, in which they receive a floating rate and pay the fixed rate

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An Example (cont’d)

On net, XYZ pays 6.9548%

XYZ net payment = – LIBOR + LIBOR – 6.9548% = –6.9548%

Floating Payment Swap Payment

Questions to Ask

Where does 6.9548% come from? How to determine this swap rate?

As a counterparty to the swap, the market-maker receives fixed and pays floating. Thus, the market-maker is facing the risk of high floating rate. (The XYZ uses interest rate swap to transfer the risk to the swap counterparty.) • The market-maker will hedge the floating rate

payments by using, for example, forward rate agreements

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Hedge Swap Position

The interest rate for year 0 to year 1 is 6%. Forward rate for year 1 to year 2 is 7.0024% and for year 2 to year 3 is 8.0071%.

Cash Flow Table

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Computing the Swap Rate

Suppose there are n swap settlements, occurring on dates ti, i = 1,… , n. Swap rate

is R. The implied forward interest rate from date ti-1

to date ti, known at date 0, is r0(ti-1, ti)

The price of a zero-coupon bond maturing on date ti is P(0, ti)

Using FRAs to hedge or using swap to hedge same present value n

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Computing the Swap Rate (cont’d)

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Thus, the fixed swap rate is as a weighted average of the implied forward rates, where zero-coupon bond prices are used to determine the weights

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Computing the Swap Rate (cont’d)

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An alternative way to express the swap rate is

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The Swap CurveA set of swap rates at different maturities

• The swap curve should be consistent with the interest rate curve implied by the Eurodollar futures contract, which is used to hedge swaps

Computing swap rates

• The Eurodollar futures contract provides a set of 3-month forward LIBOR rates

• In turn, zero-coupon bond prices can be constructed from implied forward rates

• We can then use formula to compute swap rates

The Swap Curve (cont’d)

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Maturity Date of Eurodollar Futures Contract

Price of Eurodollar Futures

3-Month Forward Rate Implied by Eurodollar Futures Price

Implied Dec 2007 Price of $1 Paid 3-Months after Futures Mat. Date

Swap Rate (%) for Loan Made Dec 2007, Ending 3 Months after Futures Mat. Date

Dec-07 95.250 0.01201 0.98814 4.8028Mar-08 95.720 0.01082 0.97756 4.5664Jun-08 95.965 0.01020 0.96769 4.4059Sep-08 96.075 0.00992 0.95818 4.2982Dec-08 96.080 0.00991 0.94878 4.2326

Table Three-month LIBOR forward rates and swap rates implied by Eurodollar futures prices with maturity dates given in the first column. Prices are from November 8, 2007. Source: Wall Street Journal online.

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Variation of Interest Rate Swaps

A deferred swap is a swap that begins at some date in the future, but its swap rate is agreed upon today An amortizing swap is a swap where the notional value is declining over time (e.g., floating rate mortgage)

An accreting swap is a swap where the notional value is growing over time

General formula for the swap rate:

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Currency Swaps

A currency swap entails an exchange of payments in different currencies

A currency swap is equivalent to borrowing in one currency and lending in another

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Currency Swap: An ExampleA dollar-based firm has a 3-year 3.5% euro-dominated bond with a €100 par. Current exchange rate is $0.90/€.

• Use currency forward contracts to hedge

Unhedged Forward HedgedYear Euro cash flow Exchange rate Dollar Cash Flow

1 -€3.5 0.9217 -$3.226

2 -€3.5 0.9440 -$3.304

3 -€103.5 0.9668 -$100.064

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Currency Swap Example (cont’d)

Alternatively, the firm can enter into a currency swap with the market-maker –– making payments on a dollar-based bond and receiving payments for its euro-based bond

Rule: the present value of the payments (from and to the market-maker) should be the same!

Firm Market-maker

$, $, $ (how much?)

€3.5, €3.5, €103.5

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Currency Swap Example (cont’d)

The euro-based par bond has value €100, which is equivalent to $90, given the current exchange rate of $0.90/€.

Therefore, the dollar-based par bond should have value $90.

Suppose the effective annual dollar-denominated interest rate is 6%

The payments on dollar-based bond are:

$5.40, $5.40, and $95.40.

Hedged or Unhedged Cash Flows

Unhedged cash flows and hedged cash flows using either swap or forward contracts.

Unhedged €3.5 €3.5€103.5

Swap-hedged $5.4 $5.4 $95.4

Forward- $3.226 $3.304 $100.064

hedged

All have PV = €100 = $90 6-36

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Currency Swap

A currency swap is equivalent to borrowing in one currency and lending in another

Firm Market-maker

$5.4, $5.4, $90

€3.5, €3.5, €103.5

$90 now

€100 now

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Currency Swap: Market-MakerMarket-maker use currency forward contracts to hedge the Euro interest. The position of the market-maker is summarized below

The PV of the market-maker’s net cash flows is($2.174 / 1.06) + ($2.096 / 1.062) – ($4.664 / 1.063) = 0

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Redundant Information?

Current exchange rate x0 is $0.90/€.

Interest rate on euro is 3.5%.

Interest rate on dollar is 6%.

Question: what determines the forward exchange rates?

Forward exchange rates are given as 0.9217, 0.9440, and 0.9668.

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Currency Swap Formulas

Consider a swap in which a dollar annuity, R, is exchanged for an annuity in another currency, R*

• There are n payments

• The time-0 forward price for a unit of foreign currency delivered at time ti is F0,ti

.

• The dollar-denominated zero-coupon bond price is P0,ti

• Given R*, what is R?

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Currency Swap Formulas (cont’d)

The PV of the two annuities must be the same (in one currency)

• Then,

This equation is equivalent to previous formula, with the implied forward rate, r0(ti-1, ti), replaced by the foreign-currency-denominated annuity payment translated into dollars, R*F0,ti

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Swap Bank

Swap bank is a financial institution that acts as an intermediary for interest and currency swaps. • Function: to find counterparties for those who

want to participate in swap agreements.

• The swap bank typically earns a slight premium for facilitating the swap.

In general, companies do not directly approach other companies in an attempt to create swap agreements. In most cases, companies don't even know the identities of their swap counterparties. 6-42

Swap Bank: ExampleBoth company A and company B need to take $5m loan. Company A prefers to pay variable rate of interest while company B prefers to pay fixed rate of interest.

Company A is big, well-known, and well-established. It is offered with 5% fixed rate or LIBOR by bank X.

Company B is less well-known and smaller. It is offered with 8% fixed rate or LIBOR+1% by bank Y.

How can a swap bank help here?6-43

Swap Bank: Example (cont’d)

The swap bank offers a swap to company A as follows:

By taking the loan from bank X at 5% and signing the swap above with the swap bank, effectively, company A is paying at a variable rate of LIBOR-0.5%. Great!

Co. A Swap Bank

LIBOR

5.5%

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Swap Bank: Example (cont’d)

The swap bank offers a swap to company B as follows:

By taking the loan from bank Y at LIBOR+1% and signing the swap above with the swap bank, efectively, company B is paying at a fixed rate of 7%. Great!

Co. B Swap Bank

6%

LIBOR

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Swap Bank: Example (cont’d)

Both company A and company B are better off.

The swap bank earns 0.5%

Concerns: default risk (credit risk) …

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LIBOR

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6%

LIBOR

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End of the Notes!

Appendix:

More on Swaps

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Swaptions

A swaption is an option (right) to enter into a swap with pre-specified terms

• Swaption can be used to speculate on the swap price in the future

A swaption is analogous to an ordinary option.

• Swaption has a premium.

• Swaptions can be American or European.

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Payer / Receiver SwaptionA payer swaption gives its holder the right, but not the obligation, to pay the pre-specified price (strike price) and receive the market swap price

• The holder of a payer swaption would exercise when the market swap price is above the strike

A receiver swaption gives its holder the right, but not the obligation to pay the market swap price and receive the pre-specified strike price

• The holder of a receiver swaption would exercise when the market swap price is below the strike

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Example: Payer Swaption

Suppose we enter into a 3-month European payer oil swaption: the strike price = $21 and the underlying swap commences in 1 year and has 2 settlements

After 3 months, the fixed price on the underlying swap is $21.50: Exercise the option, obligating us to pay $21/barrel for 2 years and allowing us to receive $21.5/barrel for 2 years.

• In year 1 and year 2, we will receive $21.50 and pay $21, for a certain net cash flow each year of $0.50

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Total Return Swaps

A total return swap is a swap, in which one party pays the realized total return (dividends plus capital gains) on a reference asset, and the other party pays a floating return such as LIBOR

The two parties exchange only the difference between these rates

The party paying the return on the reference asset is the total return payer

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Example: Total Return Swap

ABC Asset Management want to sell $1 billion of investment in S&P index

An alternative is to swap the total stock return into a floating short-term rate

Table Illustration of cash flows on a total return swap with annual settlement for 3 years.

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Total Return Swaps (cont’d)

Why to use a total return swap?

• The total return payer gives up the possible risk premium on the stock index

• The payoff for the swap is equivalent to direct selling of the stock and buying a floating-rate note

• However, the total return swap can allow foreign investors to own stocks without physically holding them, so as to avoid withholding foreign taxes

• Flexible management of credit risk