chance/brooksan introduction to derivatives and risk management, 7th ed.ch. 12: 1 chapter 12: swaps...

Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.

Ch. 12: 1

Chapter 12: Swaps

Markets are an evolving ecology. New risks arise all the Markets are an evolving ecology. New risks arise all the time.time.

Andrew LoAndrew Lo

CFA MagazineCFA Magazine, March-April, 2004, p. 31, March-April, 2004, p. 31

Ch. 12: 2

Important Concepts

The concept of a swapThe concept of a swap Different types of swaps, based on underlying currency, Different types of swaps, based on underlying currency,

interest rate, or equityinterest rate, or equity Pricing and valuation of swapsPricing and valuation of swaps Strategies using swapsStrategies using swaps

Ch. 12: 3

Nature of SwapsNature of Swaps

A swap is an agreement to exchange A swap is an agreement to exchange cash flows at specified future times cash flows at specified future times according to certain specified rules.according to certain specified rules.

Ch. 12: 4

Four types of swapsFour types of swaps CurrencyCurrency Interest rateInterest rate EquityEquity Commodity Commodity

Characteristics of swapsCharacteristics of swaps No cash up frontNo cash up front Notional principalNotional principal Settlement date, settlement periodSettlement date, settlement period Credit riskCredit risk Dealer marketDealer market

See See Figure 12.1, p. 407Figure 12.1, p. 407 for growth in world-wide notional principal for growth in world-wide notional principal

Ch. 12: 5

Interest Rate Swaps

In an interest rate swap, two parties agree to In an interest rate swap, two parties agree to exchange or swap a series of interest payments.exchange or swap a series of interest payments.

In a “plain vanilla” interest rate swap, one party In a “plain vanilla” interest rate swap, one party agrees to make a series of fixed interest payments agrees to make a series of fixed interest payments and the other agrees to make a series of variable or and the other agrees to make a series of variable or floating interest payments.floating interest payments.

Chance/Brooks 6

Example of a “Plain Vanilla” Swap

An agreement by XYZ Corp to receive 6-month LIBOR & pay a fixed rate of 5% per annum every 6 months for 3 years on a notional principal of $100 million.

Chance/Brooks 7

---------Millions of Dollars---------

LIBOR FLOATING FIXED Net

Date Rate Cash Flow Cash Flow Cash Flow

Mar.5, 2004 4.2%

Sept. 5, 2004 4.8% +2.10 –2.50 –0.40

Mar.5, 2005 5.3% +2.40 –2.50 –0.10

Sept. 5, 2005 5.5% +2.65 –2.50 +0.15

Mar.5, 2006 5.6% +2.75 –2.50 +0.25

Sept. 5, 2006 5.9% +2.80 –2.50 +0.30

Mar.5, 2007 6.4% +2.95 –2.50 +0.45

Cash Flows to XYZ Corp

Chance/Brooks 8

Uses of an Interest Rate Swap

Converting a liability from fixed rate to floating rate floating rate to fixed rate

Converting an investment from fixed rate to floating rate floating rate to fixed rate

Chance/Brooks 9

Chance/Brooks 10

Transforming a Liability

ABC XYZ

LIBOR+0.1%LIBOR+0.1%

5.2%5.2%

Chance/Brooks 11

When a Financial Institution is Involved

LIBOR LIBORLIBOR+0.1%LIBOR+0.1%

4.985% 5.015%

5.2%5.2%ABC XYZ

Chance/Brooks 12

Chance/Brooks 13

Chance/Brooks 14

Chance/Brooks 15

Transforming an Asset

ABC XYZ

LIBOR-0.2%LIBOR-0.2%

4.7%4.7%

Chance/Brooks 16

ABC F.I. XYZ

LIBOR LIBOR

4.7%4.7%

5.015%4.985%

LIBOR-0.2%LIBOR-0.2%

Chance/Brooks 17

Quotes By a Swap Dealer

Maturity Bid (%) Offer (%) Swap Rate (%)

2 years 6.03 6.06 6.045

3 years 6.21 6.24 6.225

4 years 6.35 6.39 6.370

5 years 6.47 6.51 6.490

7 years 6.65 6.68 6.665

10 years 6.83 6.87 6.850

Chance/Brooks 18

The Comparative Advantage Argument

PQR Corp wants to borrow floating RST Corp wants to borrow fixed

Fixed Floating

PQR Corp 4.00% 6-month LIBOR + 0.30%

RST Corp 5.20% 6-month LIBOR + 1.00%

Chance/Brooks 19

Chance/Brooks 20

The Swap

PQR RST

LIBOR+1%LIBOR+1%

Chance/Brooks 21

PQR F.I. RST4%4%

LIBOR LIBOR

LIBOR+1%LIBOR+1%

3.93% 3.97%

Chance/Brooks 22

Criticism of the Comparative Advantage Argument

The 4.0% and 5.2% rates available to PQR Corp and RST Corp in fixed rate markets are 5-year rates.

The LIBOR+0.3% and LIBOR+1% rates available in the floating rate market are six-month rates.

RST Corp’s fixed rate depends on the spread above LIBOR it borrows at in the future.

Chance/Brooks 23

The Nature of Swap Rates

Six-month LIBOR is a short-term AA borrowing rate.

The 5-year swap rate has a risk corresponding to the situation where 10 six-month loans are made to AA borrowers at LIBOR.

This is because the lender can enter into a swap where income from the LIBOR loans is exchanged for the 5-year swap rate.

Ch. 12: 24

Interest Rate Swaps

The Structure of a Typical Interest Rate SwapThe Structure of a Typical Interest Rate Swap Example: On December 15 XYZ enters into $50 Example: On December 15 XYZ enters into $50

million NP swap with ABSwaps. Payments will be on million NP swap with ABSwaps. Payments will be on 1515thth of March, June, September, December for one of March, June, September, December for one year, based on LIBOR. XYZ will pay 7.5% fixed and year, based on LIBOR. XYZ will pay 7.5% fixed and ABSwaps will pay LIBOR. Interest based on exact day ABSwaps will pay LIBOR. Interest based on exact day count and 360 days (30 per month). In general the cash count and 360 days (30 per month). In general the cash flow to the fixed payer will beflow to the fixed payer will be

365or 360

Daysrate) Fixed - (LIBORprincipal) (Notional

Ch. 12: 25

Interest Rate Swaps

The Structure of a Typical Interest Rate Swap The Structure of a Typical Interest Rate Swap (continued)(continued) The payments in this swap areThe payments in this swap are

Payments are netted.Payments are netted. See See Figure 12.2, p. 409Figure 12.2, p. 409 for payment pattern for payment pattern See See Table 12.1, p. 410Table 12.1, p. 410 for sample of payments after- for sample of payments after-

the-fact.the-fact.

Days0.075) - 00)(LIBOR($50,000,0

Chance/Brooks 26

Zero Rates

A zero rate (or spot rate), for maturity T is the rate of interest earned on an investment that provides a payoff only at time T

Chance/Brooks 27

Example

Maturity(years)

Zero Rate(% cont comp)

0.5 5.0

1.0 5.8

1.5 6.4

2.0 6.8

Chance/Brooks 28

Forward Rates

The forward rate is the future zero rate

implied by today’s term structure of interest rates

Chance/Brooks 29

Formula for Forward Rates

Suppose that the zero rates for time periods T1 and T2 are R1 and R2 with both rates continuously compounded.

The forward rate for the period between times T1 and T2 is

R T R T

T T2 2 1 1

Chance/Brooks 30

Calculation of Forward Rates

Zero Rate for Forward Rate

an n -year Investment for n th Year

Year (n ) (% per annum) (% per annum)

2 4.0 5.0

3 4.6 5.8

4 5.0 6.2

5 5.3 6.5

Chance/Brooks 31

Forward Rate Agreement

Forward Rate Agreement (FRA) is an agreement where interest at a predetermined rate, RK is exchanged for interest at the market rate.

An FRA can be valued by assuming that the forward interest rate is certain to be realized.

Chance/Brooks 32

FRA Valuation

Value of FRA where a fixed rate RK will be received on a principal L between times T1 and T2 is

Value of FRA where a fixed rate is paid is

RF is the forward rate for the period and R2 is the zero rate for maturity T2

22))(( 12TR

FK eTTRRL

22))(( 12TR

KF eTTRRL

Chance/Brooks 33

Example: FRA ValuationSuppose that the three-month LIBOR rate is 5% and the six-month LIBOR rate is 5.5% with continuous compounding. Consider an FRA where you will receive a rate 7% measured with quarterly compounding, on a principal of $1 million between the end of month 3 and the end of month 6. The forward rate is 6% percent with continuous compounding or 6.0452 with quarterly compounding. The value of the FRA is$1,000,000 x (.07– .060452) x 0.25 x e-0.055 x 0.5 = $2,322

Chance/Brooks 34

Valuation of an Outstanding Interest Rate Swap

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

Interest rate swaps can be valued as the difference between the value of a fixed-rate bond (Bfix) and the value of a floating-rate bond (Bfl).

Alternatively, they can be valued as a portfolio of FRAs.

Chance/Brooks 35

Valuation in Terms of Bonds

The fixed rate bond is valued in the usual way as present value of future cash flows.

The floating rate bond is valued by noting that it is worth par immediately after the next payment date.

Then the value of the swap (VSWAP) is

VSWAP = Bfix - Bfl

Chance/Brooks 36

ExampleSuppose that PDQ Corp pays six-month LIBOR and receives 8% per annum (with semiannual compounding) on a swap with notional principal of $100 million and the remaining payment are in 3, 9, and 15 months. The swap has a remaining life of 15 months. The LIBOR rates with continuous compounding for 3-month, 9-month, and 15-month maturities are 10%, 10.5%, and 11%. The 6-month LIBOR rate at the last payment date was 10.2% (with semiannual compounding). What is the value of the swap?

Chance/Brooks 37

Swap Valuation

Bfix = 4e-0.1x3/12 + 4e-0.105x9/12 + 104e-0.11x15/12

= $98.24 million

Bfl = 5.1e-0.1x3/12 + 100e -0.1x3/12

= $102.51 million

The value of the swap is

VSWAP = $98.24 million - $102.51 million

= - $4.27 million

If PDQ Corp had been paying fixed and receiving floating, the value of the swap would be + $4.27 million.

Chance/Brooks 38

Valuation in Terms of FRAs Each exchange of payments in an interest rate

swap is an FRA. The FRAs can be valued on the assumption that

today’s forward rates are realized. The procedure is as follows:

Calculate forward rates for each of the LIBOR rates that will determine swap cash flows.

Calculate swap cash flows assuming that the LIBOR rates will equal the forward rates.

Set the swap value equal to the present value of these cash flows.

Chance/Brooks 39

Swap Valuation as FRAsConsider again the situation in the previous example. The cash flows that will be exchanged in 3 months have already been determined. A rate of 8% will be exchanged for 10.2%. The value of the exchange to PDQ Corp is

0.5 x 100 x (0.08 - 0.102) e-0.1 x 3/12 = -1.07To calculate the value of the exchange in 9 months, we first calculate the forward rate corresponding to the period between 3 and 9 months:

[.105 x .75 –.10 x .25]/.5 =.1075 or 10.75%.

Chance/Brooks 40

Swap Valuation as FRAs

Using the equation

Rm = m(eRc/m – 1)

where Rc is the rate of interest with continuous compounding and Rm is the equivalent rate with compounding m times per annum, we convert 10.75% with continuous compounding into 2 x (e.1075/2 – 1) = .11044 or 11.044% with semiannual compounding.

Chance/Brooks 41

The value of the FRA corresponding to the exchange in 9 months is therefore0.5 x 100 x (0.08 - 0.11044) e-0.105 x 9/12 = -1.41

To calculate the value of the exchange in 15 months, we first calculate the forward rate corresponding to the period between 9 and 15 months. This is

[.11 x 1.25 –.105 x .75]/.5 =.1175 or 11.75%.This value becomes 12.102% with semiannual compounding.

Chance/Brooks 42

The value of the FRA corresponding to the exchange in 15 months is therefore

0.5 x 100 x (0.08 - 0.12102) e-0.11 x 15/12 = -1.79

The total value of the swap is

-1.07 – 1.41 – 1.79 = -$4.27 million

Ch. 12: 43

Interest Rate Swaps

Interest Rate Swap StrategiesInterest Rate Swap Strategies See See Figure 12.5, p. 418Figure 12.5, p. 418 for example of converting for example of converting

floating-rate loan into fixed-rate loanfloating-rate loan into fixed-rate loan Other types of swapsOther types of swaps

• Index amortizing swapsIndex amortizing swaps

• Diff swapsDiff swaps

• Constant maturity swapsConstant maturity swaps

Ch. 12: 44

Currency Swaps

In a currency swap, the parties make either fixed or In a currency swap, the parties make either fixed or variable payments to each other in different currencies.variable payments to each other in different currencies.

Example: Reston Technology enters into currency swap Example: Reston Technology enters into currency swap with GSI. Reston will pay euros at 4.35% based on NP of with GSI. Reston will pay euros at 4.35% based on NP of €€10 million semiannually for two years. GSI will pay 10 million semiannually for two years. GSI will pay dollars at 6.1% based on NP of $9.804 million dollars at 6.1% based on NP of $9.804 million semiannually for two years. semiannually for two years. See See Figure 12.6, p. 421Figure 12.6, p. 421..

Note the relationship between interest rate and currency Note the relationship between interest rate and currency swaps in swaps in Figure 12.7, p. 422Figure 12.7, p. 422..

Chance/Brooks 45

Exchange of Principal

In an interest rate swap the principal is not exchanged.

In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life.

Chance/Brooks 46

An Example of a Currency Swap

An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years.

This is a fixed-for-fixed currency swap.

Chance/Brooks 47

The Cash Flows

Dollars Pounds$

------millions------

2004 –15.00 +10.002005 +1.20 – 1.10

2006 + 1.20 – 1.10 2007 + 1.20 – 1.10

2008 + 1.20 – 1.10 2009 +16.20 −11.10

Chance/Brooks 48

Typical Uses of a Currency Swap

Conversion from a liability in one currency to a liability in another currency.

Conversion from an investment in one currency to an investment in another currency.

Comparative Advantage Arguments for Currency Swaps

Shell wants to borrow UK £

BP wants to borrow US $

US $ UK £

Shell 5.0% 12.6%

BP 7.0% 13.0%

Chance/Brooks 50

Chance/Brooks 51

Financial Institution is Involved

£ 11.9% £ 13.0%£ 13.0%

$ 5.0% $ 6.3%

$ 5.0%Shell BP

Chance/Brooks 52

BP Bears FX Risk

£ 11.9% £ 11.9%£ 13.0%

$ 5.0% $ 5.2%

$ 5.0%Shell BP

Chance/Brooks 53

Shell Bears FX Risk

£ 13.0% £ 13.0%£ 13.0%

$ 6.1% $ 6.3%

$ 5.0%Shell BP

Chance/Brooks 54

Valuation of Currency Swaps Like interest rate swaps, currency swaps can be

valued either as the difference between two bonds or as a portfolio of forward contracts.

If we define VSWAP as the value in US dollars of aswap where dollars are received and a foreign currency is paid, then

VSWAP = BD – S0BF

where BF is the value, measured in foreign currency, of the foreign-denominated bond underlying the swap, BD is the value of the US dollar bond underlying the swap, and S0 is the spot exchange rate (expressed as number of units of domestic currency per unit of foreign currency).

Chance/Brooks 55

Example

Suppose that the term structure of interest rate is flat in both Japan and US at 4% and 9% per annum, respectively (both with continuous compounding). ABM Corp has entered into a currency swap to receive 5% per annum in yen and pay 8% per annum in dollars once a year. The principals in two currencies are $10 million and 1,200 million yen. The swap will last for another three years, and the current exchange is 110 yen = $1. What is the value of the swap in dollars?

Chance/Brooks 56

Swap ValuationBD = 0.8e-0.09x1 + 0.8e-0.09x2 + 10.8e-0.09x3

= 9.644 million dollars

BF = 60e-0.04x1 + 60e-0.04x2 + 1,260e-0.04x3

= 1,230.55 million yen

The value of the swap in dollars is

1,230.55/110 – 9.644 = $1.543 million

If ABM Corp had been paying yen and receiving dollars, the value of the swap would have been -$1.543 million.

Chance/Brooks 57

Currency Swap as FRAsConsider the situation in the previous example. The current spot rate is 110 yen per dollar, or 0.009091 dollar per yen. Using the equation

F0 = S0e (r - rf)T

we calculate the one-year, two-year, and three-year forward rates as

0.009091e(.09 - .04) x 1 = 0.0095570.009091e(.09 - .04) x 2 = 0.0100470.009091e(.09 - .04) x 3 = 0.010562

Chance/Brooks 58

Currency Swap as FRAs

The exchange of interest involves receiving 60 million yen and paying $0.8 million. The risk-free interest rate in dollars is 9% per annum. The value of the forward contracts corresponding to these exchanges are as follows:

(60 x 0.009557 – 0.8)e -0.09 x 1 = -0.2071

(60 x 0.010047 – 0.8)e -0.09 x 2 = -0.1647

(60 x 0.010562 – 0.8)e -0.09 x 3 = -0.1269

Chance/Brooks 59

Currency Swap as FRAs

The final exchange of principal involves receiving 1,200 million yen and paying $10 million. The value of the forward contract corresponding to the exchange is

(1,200 x 0.010562 – 10)e -0.09 x 3 = 2.0416

The total value of the swap is

2.0416 – 0.1269 – 0.1647 – 0.2071

= $1.543 million

Ch. 12: 60

Currency Swaps

Currency Swap StrategiesCurrency Swap Strategies A typical case is a firm borrowing in one currency A typical case is a firm borrowing in one currency

and wanting to borrow in another. See and wanting to borrow in another. See Figure 12.8, p. 429Figure 12.8, p. 429 for Reston-GSI example. for Reston-GSI example. Reston could get a better rate due to its familiarity Reston could get a better rate due to its familiarity to GSI and also due to credit risk.to GSI and also due to credit risk.

Also a currency swap be used to convert a stream of Also a currency swap be used to convert a stream of foreign cash flows. This type of swap would foreign cash flows. This type of swap would probably have no exchange of notional principals.probably have no exchange of notional principals.

Ch. 12: 61

Ch. 12: 62

Equity Swaps

In an equity swap, at least one of the two parties makes In an equity swap, at least one of the two parties makes payments determined by the price of a stock, the value of a payments determined by the price of a stock, the value of a stock portfolio, or the level of a stock index.stock portfolio, or the level of a stock index.

The other party’s payment can be determined by another The other party’s payment can be determined by another stock, portfolio or index, or by an interest rate, or it can be stock, portfolio or index, or by an interest rate, or it can be fixed.fixed.

CharacteristicsCharacteristics One party pays the return on an equity, the other pays One party pays the return on an equity, the other pays

fixed, floating, or the return on another equityfixed, floating, or the return on another equity Rate of return is paid, so payment can be negativeRate of return is paid, so payment can be negative Payment is not determined until end of periodPayment is not determined until end of period

Ch. 12: 63

Equity Swaps

The Structure of a Typical Equity SwapThe Structure of a Typical Equity Swap Cash flow to party paying stock and receiving fixedCash flow to party paying stock and receiving fixed

Example: IVM enters into a swap with FNS to pay Example: IVM enters into a swap with FNS to pay S&P 500 Total Return and receive a fixed rate of S&P 500 Total Return and receive a fixed rate of 3.45%. The index starts at 2710.55. Payments 3.45%. The index starts at 2710.55. Payments every 90 days for one year. Net payment will be every 90 days for one year. Net payment will be

period settlementover stock on Return

365or 360

Daysrate) (Fixed

principal) (Notional

period settlementover index stock on Return 360

90.034500)($25,000,0

Ch. 12: 64

Equity Swaps

The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued) The fixed payment will beThe fixed payment will be

• $25,000,000(.0345)(90/360) = $215,625$25,000,000(.0345)(90/360) = $215,625 See See Table 12.8, p. 431Table 12.8, p. 431 for example of payments. for example of payments.

The first equity payment isThe first equity payment is

So the first net payment is IVM pays $285,657.So the first net payment is IVM pays $285,657.

282,501$12710.55

2764.900$25,000,00

Ch. 12: 65

Equity Swaps The Structure of a Typical Equity Swap (continued)The Structure of a Typical Equity Swap (continued)

If IVM had received floating, the payoff formula If IVM had received floating, the payoff formula would bewould be

If the swap were structured so that IVM pays the If the swap were structured so that IVM pays the return on one stock index and receives the return on return on one stock index and receives the return on another, the payoff formula would beanother, the payoff formula would be

period settlementover stock on Return

Days(LIBOR)

principal) (Notional

indexstock other on Return -index stock oneon Return principal) (Notional

Ch. 12: 66

Equity Swaps Equity Swap StrategiesEquity Swap Strategies

Used to synthetically buy or sell stockUsed to synthetically buy or sell stock See See Figure 12.9, p. 437Figure 12.9, p. 437 for example. for example. Some risksSome risks

• defaultdefault

• tracking errortracking error

• cash flow shortagescash flow shortages

Ch. 12: 67

Synthetic Trading of a Portfolio

Currently the investor owns a portfolio of S&P 500 stocks Currently the investor owns a portfolio of S&P 500 stocks worth $1M.worth $1M.

Sell the portfolio for $1M and reinvest in Treasury for 3 Sell the portfolio for $1M and reinvest in Treasury for 3 months.months.

After 3 months: S&P 500 return : -1.77%; Treasury: 1.23%.After 3 months: S&P 500 return : -1.77%; Treasury: 1.23%. Buy back S&P 500 stocks portfolio.Buy back S&P 500 stocks portfolio. Alternatively, the investor can enter into an equity swap to Alternatively, the investor can enter into an equity swap to

receive Treasury rate and pay S&P 500 return for 3 monthsreceive Treasury rate and pay S&P 500 return for 3 months without selling and buying back the portfolio without selling and buying back the portfolio thus saving round trip transaction cost. thus saving round trip transaction cost.

Ch. 12: 68

Some Final Words About Swaps Similarities to forwards and futuresSimilarities to forwards and futures Offsetting swapsOffsetting swaps

Go back to dealerGo back to dealer Offset with another counterpartyOffset with another counterparty Forward contract or option on the swapForward contract or option on the swap

Chance/Brooks 69

Swaps & Forwards

A swap can be regarded as a convenient way of packaging forward contracts.

The “plain vanilla” interest rate swap consists of a series of FRAs.

The “fixed for fixed” currency swap in consists of a cash transaction & a series of forward contracts.

Chance/Brooks 70

Swaps & Forwards

The value of the swap is the sum of the values of the forward contracts underlying the swap.

Swaps are normally “at the money” initially This means that it costs nothing to enter

into a swap. It does not mean that each forward

contract underlying a swap is “at the money” initially.

Chance/Brooks 71

Credit Risk

A swap is worth zero to a company initially.

At a future time its value is liable to be either positive or negative.

The company has credit risk exposure only when its value is positive.

Chance/Brooks 72

Other Types of Swaps

Floating-for-floating interest rate swaps, amortizing swaps, step up swaps, forward swaps, constant maturity swaps, compounding swaps, LIBOR-in-arrears swaps, accrual swaps, diff swaps, cross currency interest rate swaps, equity swaps, extendable swaps, puttable swaps, swaptions, commodity swaps, volatility swaps……..

Ch. 12: 73

(Return to text slide)

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