Interest Rate and Currency Swaps

Read Chapter 7

Outline

  • Interest Rate SWAPS
  • Comparative Advantage
  • Bootstrapping
  • Valuing an Interest-Rate SWAP
  • Currency SWAP
  • Overnight Indexed Swaps
  • Other SWAP Types

Interest Rate SWAPS

  • A swap is where two parties agree to exchange cash flows at specified future times according to certain specified rules
  • An Example
    • Microsoft enters an agreement to receive 6-month LIBOR and pay a fixed rate of 4% per annum every 6 months for 3 years on a notional principal of $100 million
    • Notional principal – parties do not exchange principal; only to calculate interest payments
    • Called "Plain vanilla" - just plain ole vanilla ice cream
    • We do not consider the day count conventions such as 360, actual, etc.
    • The previous Libor rate determines the current floating rate
    • Microsoft's cash flows below
Date Libor Floating Rate
(%)
Floating Cash Flow
($ million)
Fixed Cash Flow
($ million)
Net Cash Flow
($ million)
Feb 10, 2014 3.50      
Aug 10, 2014 3.80 +1.75 2.00 0.25
Feb 10, 2015 4.00 +1.90 2.00 0.10
Aug 10, 2015 3.80 +2.00 2.00 0.00
Feb 10, 2016 4.05 +1.90 2.00 0.10
Aug 10, 2016 4.20 +2.025 2.00 +0.025
Feb 10, 2017 4.30 +2.10 2.00 +0.10
  • Using an interest rate swap
    • Convert a liability from
      • Fixed rate to floating rate
      • Floating rate to fixed rate
    • Convert an investment from
      • Fixed rate to floating rate
      • Floating rate to fixed rate
  • Example - Microsoft
    • Microsoft could have a variable-rate loan
    • Microsoft can convert it to a fixed rate loan
    • Enter a swap and receive floating
      • The floating cancels the cash flows if Microsoft has another variable-rate loan
      • Pay fixed
      • Microsoft must believe interest rates will rise
    • Swap does not eliminate the original loans
    • Swap changes the cash flows
  • SWAP
    • Ford borrows LIBOR + 0.2% from a bank
      • Enters a SWAP
        • Pays 5.5% fixed rate
        • Receives LIBOR
      • Convert a variable rate loan into a fixed loan
        • Ford borrowing cost = LIBOR + 0.2%+ 5.5% LIBOR = 5.7%
        • Ford comes out ahead on the swap if a bank would charge more than 5.7% on a fixed loan
      • Ford believes interest rates will rise
    • GM borrows at 6.0% fixed from a bank
      • Enters a SWAP
        • Receives 5.5% fixed
        • Pays LIBOR
      • Convert fixed-rate loan into variable-rate loan
        • GM borrowing cost = 6.0% + LIBOR 5.5% = LIBOR + 0.5%
        • GM comes out ahead on the swap if a bank would charge a greater rate of LIBOR +0.5% on a variable rate loan
      • GM believes interest rates will fall

Interest rate swap

  • Financial institution sets up the SWAP
    • Financial institution
      • For fixed loans:
        • Receives 5.6% and pays 5.4%
        • Profit = 5.6% – 5.4% = 0.2%
      • For variable loans:
        • Receives and pays LIBOR
        • Profit = LIBOR – LIBOR = 0%
    • Financial institution shares in the gains of a SWAP

Interest rate swap with an intermediary

  • Investment SWAP
    • Ford receives 5.0% fixed
      • Could enter a SWAP
        • Pays 5.3%
        • Receives LIBOR
      • Convert fixed-interest earning asset into variable
        • Net investment = 5% + LIBOR – 5.3% = LIBOR – 0.3%
        • Ford does better SWAP if it were offered a lower investment rate than LIBOR – 0.3%
      • Ford believes interest rates will rise
    • GM receives LIBOR – 0.5%
      • Could enter a SWAP
        • Pays LIBOR
        • Receives 5.3% fixed
      • Convert variable-interest earning asset into fixed
        • Net investment = LIBOR – 0.5% – LIBOR + 5.3% = 4.8%
        • GM does better with SWAP then if it were offered a rate lower than 4.8%
      • GM believes interest rate must fall

Interest rate swap for an investment

  • Intermediary
    • Profits from fixed leg and earns zero on bottom leg
    • profit = 5.4% – 5.2% = 0.2%
  • Intermediary can earn a loss on one of the legs
    • Net cash flows matter

Interest rate swap for an investment with intermediary

  • Day Count
    • A day count convention is specified for fixed and floating payment
    • For example, LIBOR is likely to be actual/360 in the US because LIBOR is a money market rate
  • Confirmations specify the terms of a transaction
    • The International Swaps and Derivatives has developed Master Agreements that can be used to cover all agreements between two counterparties
    • Central clearing is used for most standard swaps

Comparative Advantage

  • Airbus Corp wants to borrow floating
    • Airbus Corp borrows fixed
  • Boeing Corp wants to borrow fixed
    • Boeing Corp borrows floating
  • Parties can gain by entering a swap
  • They pay less interest rate than if borrow from bank using rates in the table
    • Comparative Advantage – companies borrow from the low cost source and exchange (swap) payments
      • Gain from Fixed: 6.3% – 5.0% = 1.30%
      • Gain from Variable: LIBOR + 0.2% – (LIBOR – 0.2%) = 0.4%
      • Total gain: 1.30% – 0.4% = 0.9%
      • Companies and bank can divide 0.9% among themselves
        • Give each company an equal benefit of 0.45%
  Fixed Floating
Airbus Corp 5.0% 6-month LIBOR – 0.2%
Boeing 6.3% 6-month LIBOR + 0.2%
  • Boeing Corp borrows LIBOR + 0.2% from bank
    • Enters SWAP
      • Pays 5.65% fixed
      • Receives LIBOR
    • Net Interest: LIBOR – (LIBOR + 0.2%) – 5.65% = –5.85%
      • Boeing lowers borrowing cost by 0.45% via the SWAP
      • The bank would charge 6.3%
  • Airbus Corp pays 5.0% fixed to bank
    • Enters SWAP
      • Pays LIBOR
      • Receives 5.65%
    • Net Interest: 5.65% – 5.0% – LIBOR = 0.65% – LIBOR = – (LIBOR – 0.65)
      • Airbus lowers borrowing costs by 0.45% via the SWAP
      • The bank would charge LIBOR – 0.2%

Interest rate swap

  • Allow a bank to earn 0.30% on SWAP
    • That leaves 0.6% for the companies
    • Each company can benefit by 0.3%
    • Verify each company receives a benefit of 0.3% while the bank earns 0.3%
  • Requires trial and error to balance interest rates

Interest rate swap

  • Criticism of the Comparative Advantage Argument
    • The 5.0% and 6.3% rates available to Airbus Corp and Boeing Corp in fixed rate markets are 5-year rates
    • The LIBOR−0.2% and LIBOR+0.2% rates available in the floating rate market are six-month rates
    • Boeing Corp's fixed rate depends on the spread above LIBOR it borrows at in the future
  • 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

Bootstrapping

  • Bootstrap the LIBOR/Swap Zero Curve when using LIBOR discounting
    • Consider a new swap where the fixed rate is the swap rate
    • When principals are added to both sides on the final payment date the swap is the exchange of a fixed rate bond for a floating rate bond
    • The floating-rate rate bond is worth par.
      • The swap is worth zero.
      • The fixed-rate bond must therefore also be worth par
    • This shows that swap rates define par yield bonds that can be used to bootstrap the LIBOR (or LIBOR/swap) zero curve
  • Example
    • The LIBOR/swap rates with continuous compounding are
      • 6-month 4.2%
      • 12-month 4.4%
      • 18-month 4.6%
    • Two-year swap rate is 6% and pays semi-annually
    • The 2-year LIBOR/swap rate, R, is 5.979%

Bootstrapping an interest rate

Valuing an Interest-Rate SWAP

  • Interest-rate SWAP
    • Initially interest rate swaps are worth close to zero
    • At later times they can be valued as a portfolio of forward rate agreements (FRAs)
  • Example
    • Receive six-month LIBOR
      • 6-month LIBOR on last payment date was 3.1%, semi-annual compounding
    • Pay 4%, semi-annual compounding, on a principal of $10 million
    • Remaining life is 1.5 years
    • LIBOR rates for 6-month, 12-month and 18-month are 3.0%, 3.5%, and 3.8%, continuous compounding
  • Method 1 – Use forward rates
    • Each exchange of payments in an interest rate swap is an FRA – fixed for variable
    • We value the FRAs assuming the forward rates are good forecasts of future rates
    • The forward rates can be calculated directly from the LIBOR/swap zero curve
    • Calculate Forward Rates
      • 6 to 12 month period: Forward rate is 4.0%, continuous
        • In semi-annual compounding, 3.96%
      • 12 to 18 month period: Forward rate is 4.4%, continuous
        • In semi-annual compounding, 4.35%
  • The table show the payoffs
    • Floating rate – just divide the semi-annual rates by 2
    • Holder receives floating and pays fixed
Time Pay Fixed
cash flow
Receive Floating
cash flow
Net Cash Flow Discount factor PV
0.5 -0.2 +0.1550 -0.0450 0.9851 -0.0443
1.0 -0.2 +0.2020 +0.0020 0.9656 +0.0019
1.5 -0.2 +0.2224 +0.0224 0.9446 +0.0212
Total         -0.0212
  • Method 2 - Valuating as if interest-rate SWAP were bonds
    • Easier method
    • The fixed rate bond is discounted as a normal cash flow
      • We act if we pay the principal in the end, $10 million
    • The floating rate bond is valued by noting that it is worth as if paid in full during the next payment date
      • We do not know the future interest rates
      • We act if we receive the principal, $10 million
    • We calculate the present value for the next payment
Time Pay fixed
cash flow
Receive variable
cash flow
Discount factor PV Fixed PV Variable
0.5 -0.2000 +10.1550 0.9851 -0.1970 +10.0038
1.0 -0.2000   0.9656 -0.1931  
1.5 -10.2000   0.9446 -9.6349  
Total       -10.0250 +10.0038

 

Swap value = 10.0038 − 10.0250 = -0.0212 million

 

Currency Swap

  • A swap can be regarded as a convenient way of packaging forward contracts
    • When a swap is initiated the swap has zero value, but typically some forwards have a positive value and some have a negative value
  • Exchange of Principal
    • For an interest rate swap, the parties do not exchange the principal
      • Called notional
    • For a currency swap, the parties exchange the principal at the beginning and the end of the swap
      • One party needs one currency while the other party needs the other currency
  • Typical Uses
    • Convert a liability in one currency to a liability in another currency
    • Convert an investment in one currency to an investment in another currency
  • Example
    • A company agrees to pay 7% on a euro principal of €20,000,000 & receive 5% on a US$ principal of $10,000,000 every year for 5 years
    • Table below shows cash flows
Date Dollar Cash Flows
(millions)
Euro cash flow
(millions)
May 1, 2012 –10.00 +20.00
May 1, 2013 +0.50 –1.40
May 1, 2014 +0.50 –1.40
May 1, 2015 +0.50 –1.40
May 1, 2016 +0.50 –1.40
May 1, 2017 +10.50 –21.40
  • Comparative Advantage
    • Could originate from taxes
      • Boeing wants to borrow Euros
      • Airbus wants to borrow USD
      • Cost after adjusting for the differential impact of taxes
  USD Euros
Boeing 6.0% 8.5%
Airbus 7.5% 9.0%
  • Two methods to value currency swaps
    • Value cash flows as two bonds that pay in different currencies
      • Use exchange rate to convert to same currency
    • Calculate the currency forward rates
      • Currency forward contains the exchange rate
  • Example
    • Current exchange rate is 1.50 USD per Euro
      • All Euro LIBOR/swap rates are 5%
      • All USD LIBOR/swap rates are 7%
    • Fixed payments are made annually
      • 6% is paid in dollars
      • 7% is received in euros
    • Principals are $30 million and €20 million
    • Swap will last for 3 more years
  • First Method – Treat cash flows as if they are bonds
    • Table shows the calculations
Time Cash Flows ($) PV ($) Cash flows (€) PV (€)
1 -1.8 -1.6783 +1.4 1.3317
2 -1.8 -1.5648 +1.4 1.2668
3 -1.8 -1.4591 +1.4 1.2050
3 -30.0 -24.3175 +20.0 17.2142
Total   -29.0197   21.0176

 

Value ($) = -$29.0197 + 21.0176€ x 1.5 $/€ = $2.507 million

  • Second Method - Use currency forwards to value SWAP
    • Calculate the currency forward rates by using interest rates
      • Equation is below
      • t is the year
    • Multiply the Euro cash flow by the currency forward to convert to US$
    • Use the U.S. interest rate to discount

Currency forward

Time Pay cash flow
($)
Receive cash flow
(€)
Currency Forward
($ per €)
Receive cash flow
($)
Net Cash Flow
($)
Present value
($)
1 -1.8 +1.4 1.5303 2.1424 0.3424 0.3193
2 1.8 +1.4 1.5612 2.1857 0.3857 0.3353
3 -1.8 +1.4 1.5928 2.2299 0.4299 0.3484
3 -30.0 +20.0 1.5928 31.8551 1.8551 1.5037
Total           2.507
  • Other Currency Swaps
    • Fixed-for-floating: equivalent to a fixed-for-fixed currency swap plus a fixed for floating interest rate swap
    • Floating-for-floating: equivalent to a fixed-for-fixed currency swap plus two floating interest rate swaps

Overnight Indexed Swaps

  • Overnight Indexed Swaps
    • Fixed rate for Overnight Indexed SWAPS (OIS)
      • Usually for three months
    • Variable interest rate
      • Geometric average of overnight interest rates during period
      • Geometric average
      • Rolling over daily, i.e. compounding daily, (1 + i1) (1 + i2) (1 + i3)∙∙∙ (1 + in)
      • Geometric average

Geometric average

  • If fixed rate exceeds variable rate, the fixed-rate payer pays to the floating-rate receiver
  • If fixed rate is less than variable rate, then the fixed-rate payer receives difference.
  • No exchange for principle
    • On maturity, calculate the geometric rate
    • Loser pays the winner the difference
  • If OIS rate = LIBOR rate?
    • A bank can
      • Borrow $100 million in the overnight market (variable)
        • Roll forward for 3 months
        • Pay federal funds rate – U.S. banks lend to other banks overnight
      • Enter a SWAP
        • Receive variable rate for OIS
          • Geometric of average
        • Pay fixed rate of OIS for 3 months
      • Lend the funds to another bank at LIBOR for 3 months
        • Note: Bank is borrowing at a variable rate and lending at a fixe rate.
        • The SWAP connects the two, so bank does not get squeezed if interest rate changes
  • The excess of LIBOR over the OIS rate is the LIBOR-OIS spread.
    • It is usually about 10 basis points but spiked at an all time high of 364 basis points in October 2008
    • 2008 Global Financial Crisis
  • OIS vs LIBOR discounting
    • Investors have traditionally used LIBOR rates (and swap rates determined from swaps where LIBOR is exchanged for fixed) as risk-free rates to value derivatives
    • Most market participants now use the OIS rate as the discount rate when collateralized deals are valued, but continue to use LIBOR rates for discounting cash flows in non-collateralized deals
  • Valuation of Swaps Using OIS discounting
    • Zero rates are bootstrapped from OIS rates
    • This is similar to the way the LIBOR/swap zero curve is produced
    • Forward LIBOR rates are then calculated so that so that swaps entered into at the current swap rate are worth zero
    • The swap is valued by assuming that forward LIBOR is realized and discounting at the OIS rate
    • There is no simple way of valuing the swap in terms of bonds

Other SWAP Types

  • 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
    • Some swaps are more likely to lead to credit risk exposure than others
    • What is the situation if early forward rates have a positive value?
    • What is the situation when the early forward rates have a negative value?
  • Credit Default Swaps:
    • A Quick First Look Notional principal (e.g. $100 million) and maturity (e.g. 5 yrs) specified
    • Usually bonds
    • Buyer pays a fixed rate (e.g. 150 bp) on the notional principal (the CDS spread)
      • Buyer is buying insurance on a security
      • If the security drops in value, the buyer can exercise CDS
    • Investment banks wrote trillions of dollars of CDS
      • One the factors that amplified the 2008 Global Financial Crisis
      • Total face value of bonds bought equals notional principal
  • Other Types of Swaps
    • Amortizing/ step up
    • Compounding swap
    • Constant maturity swap
    • LIBOR-in-arrears swap
    • Accrual swap
    • Equity swap
    • Cross currency interest rate swap
    • Floating-for-floating currency swap
    • Diff swap
    • Commodity swap
    • Variance swap
 

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