chance/brooksan introduction to derivatives and risk management, 7th ed.ch. 12: 1 chapter 12: swaps...
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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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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.
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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.
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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.
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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
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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
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Chance/Brooks 9
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Chance/Brooks 10
Transforming a Liability
ABC XYZ
LIBOR
5%
LIBOR+0.1%LIBOR+0.1%
5.2%5.2%
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Chance/Brooks 11
When a Financial Institution is Involved
F.I.
LIBOR LIBORLIBOR+0.1%LIBOR+0.1%
4.985% 5.015%
5.2%5.2%ABC XYZ
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Chance/Brooks 12
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Chance/Brooks 13
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Chance/Brooks 14
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Chance/Brooks 15
Transforming an Asset
ABC XYZ
LIBOR
5%
LIBOR-0.2%LIBOR-0.2%
4.7%4.7%
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Chance/Brooks 16
When a Financial Institution is Involved
ABC F.I. XYZ
LIBOR LIBOR
4.7%4.7%
5.015%4.985%
LIBOR-0.2%LIBOR-0.2%
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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
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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%
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Chance/Brooks 19
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Chance/Brooks 20
The Swap
PQR RST
LIBOR
LIBOR+1%LIBOR+1%
3.95%
4%4%
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Chance/Brooks 21
When a Financial Institution is Involved
PQR F.I. RST4%4%
LIBOR LIBOR
LIBOR+1%LIBOR+1%
3.93% 3.97%
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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.
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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.
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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.
360
Days0.075) - 00)(LIBOR($50,000,0
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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
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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
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Chance/Brooks 28
Forward Rates
The forward rate is the future zero rate
implied by today’s term structure of interest rates
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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
2 1
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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)
1 3.0
2 4.0 5.0
3 4.6 5.8
4 5.0 6.2
5 5.3 6.5
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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.
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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
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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
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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.
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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
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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?
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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.
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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.
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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%.
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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.
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Chance/Brooks 41
Swap Valuation as FRAs
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.
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Chance/Brooks 42
Swap Valuation as FRAs
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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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
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Chance/Brooks An Introduction to Derivatives and Risk Management, 7th ed.
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 $9.804 million semiannually for two years. semiannually for two years. See See Figure 12.6, p. 421Figure 12.6, p. 421..
Note the relationship between interest rate and currency Note the relationship between interest rate and currency swaps in swaps in Figure 12.7, p. 422Figure 12.7, p. 422..
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Exchange of Principal
In an interest rate swap the principal is not exchanged.
In a currency swap the principal is usually exchanged at the beginning and the end of the swap’s life.
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An Example of a Currency Swap
An agreement to pay 11% on a sterling principal of £10,000,000 & receive 8% on a US$ principal of $15,000,000 every year for 5 years.
This is a fixed-for-fixed currency swap.
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The Cash Flows
Year
Dollars Pounds$
------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
£
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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.
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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%
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Financial Institution is Involved
F.I.
£ 11.9% £ 13.0%£ 13.0%
$ 5.0% $ 6.3%
$ 5.0%Shell BP
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BP Bears FX Risk
F.I.
£ 11.9% £ 11.9%£ 13.0%
$ 5.0% $ 5.2%
$ 5.0%Shell BP
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Shell Bears FX Risk
F.I.
£ 13.0% £ 13.0%£ 13.0%
$ 6.1% $ 6.3%
$ 5.0%Shell BP
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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).
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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?
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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.
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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
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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
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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
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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.
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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
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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
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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
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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
360
Days(LIBOR)
principal) (Notional
indexstock other on Return -index stock oneon Return principal) (Notional
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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
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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.
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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
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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.
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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.
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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.
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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……..
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