Zvi Wiener

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Swaps

• Zvi Wiener
• 02-588-3049
• http//pluto.mscc.huji.ac.il/mswiener/zvi.html

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Interest Rate Swaps Concept

• An agreement between 2 parties to exchange
  periodic payments calculated on the basis of
  specified interest rates and a notional amount.
  
• Plain Vanilla Swap

Based on a presentation of Global Risk Strategy
Group of Deutsche Bank
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IRS

• In a standard IRS, one leg consists of fixed
  rate payments and the other depends on the
  evolution of a floating rate.
  
• Typically long dated contracts 2-30 years
• Sometimes includes options, amortization, etc.
• Interest compounded according to different
  conventions (eg 30/360, Act/Act. Act/360, etc.)

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IRS Origins

• AAA wants to borrow in floating and BBB wants to
  borrow in fixed.
  
• Fixed Floating
• AAA 7.00 LIBOR5bps
• BBB 8.50 LIBOR85bps
• difference 1.5 0.8
• Net differential 70bps 0.7

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Comparative Advantage
7.0
Libor85bp
BBB
AAA

• Cost of funds for AAALibor - 40bp (45bps saved)
• Cost of funds for BBB8.25 (25bps saved)
• Swap rate 7.40
• Swap rate is the fixed rate which is paid against
  receiving Libor.

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Basic terms of IRS

• Notional amount
• Fixed rate leg
• Floating rate leg
• Calculated period
• Day count fraction

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Basic terms of IRS

• Payer and receiver - quoted relative to fixed
  interest (i.e. payer payer of fixed rate)
  
• buyer payer, seller receiver
• Short party payer of fixed, (buyer)
• Long party receiver of fixed, (seller)
• Valuation net value NOT notional!!

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Various swaps

• Coupon swaps - fixed against floating.
• Basis or Index swaps - exchange of two streams
  both are computed using floating IR.
  
• Currency swap - interest payments are
  denominated in different currencies.
  
• Asset swap - to exchange interest received on
  specific assets.
  
• Term swap maturity more then 2 years.
• Money Market swap - less then 2 years.

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Payments

• Fixed payment
• (notional)(Fixed rate)(fixed rate day count
  convention)
  
• Floating payment
• (notional)(Float. rate)(float. rate day count
  convention)

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Time Value of Money

• present value PV CFt/(1r)t
• Future value FV CFt(1r)t
• Net present value NPV sum of all PV

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Swap Pricing

• A swap is a series of cash flows.
• An on-market swap has a Net Present Value of
  zero!
  
• PV(Fixed leg) PV(Floating leg) 0

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Pricing

• Floating leg is equal to notional amount at each
  day of interest rate settlement (by definition of
  LIBOR).
  
• Fixed leg can be valued by standard NPV, since
  the paid amount is known.

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Forward starting swaps

• interest starts accruing at some date in the
  future.
  
• Valuation is similar to a long swap long and a
  short swap short.

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• Zero coupon swap (reinvested payments)
• Amortizing swap (decreasing notional)
• Accreting swap (increasing notional)
• Rollercoaster (variable notional)

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Amortizing swap
Decreasing notional affects coupon payments
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Unwinding an existing swap

• Enter into an offsetting swap at the prevailing
  market rate.
  
• If we are between two reset dates the offsetting
  swap will have a short first period to account
  for accrued interest.
  
• It is important that floating payment dates
  match!!

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Unwinding
Net of the two offsetting swaps is 2 for the
life of the contract. (sometimes novation)
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Risks of Swaps

• Interest rate risk - value of fixed side may
  change
  
• Credit risk - default or change of rating of
  counterparty
  
• Mismatch risk - payment dates of fixed and
  floating side are not necessarily the same
  
• Basis risk and Settlement risk

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Credit risk of a swap contract

• Default of counterparty (change of rating).
• Exists when the value of swap is positive
• Frequency of payments reduces the credit risk,
• similar to mark to market.
• Netting agreements.
• Credit exposure changes during the life of a swap.

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Duration of a swap

• Fixed leg has a long duration (approximately).
• Short leg has duration about time to reset.
• Duration is a measure of price sencitivity to
  interest rate changes (approximately is equal to
  average time to payment).

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IRS Markets

• Daily average volume of trade (notional)
• 1995 1998 2001
• 63B 155B 331B

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Mark to market

• daily repricing
• collateral
• adjustments
• reduces credit exposure

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Reasons to use swaps by firms

• Lower cost of funds
• Home market effects
• Comparative advantage of highly rated firms

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FRM-GARP 0047

• Which one of the following deals has the largest
  credit exposure for a 1,000,000 deal size.
  Assume that the counterparty in each deal is a
  AAA-rated bank and there is no settlement risk.
• A. Pay fixed in an interest rate swap for 1 year
• B. Sell USD against DEM in a 1 year forward
  contract.
  
• C. Sell a 1-year DEM Cap
• D. Purchase a 1-year Certificate of Deposit

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FRM-GARP 0047

• Which one of the following deals has the largest
  credit exposure for a 1,000,000 deal size.
  Assume that the counterparty in each deal is a
  AAA-rated bank and there is no settlement risk.
• A. Pay fixed in an interest rate swap for 1 year
• B. Sell USD against DEM in a 1 year forward
  contract.
  
• C. Sell a 1-year DEM Cap
• D. Purchase a 1-year Certificate of Deposit

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Global Derivatives Markets 1999
OTC Instruments 88T
Exchange traded 13.5T

• IR contracts 60,091
• FRAs 6,775
• Swaps 43,936
• Options 9,380
• FX contracts 14,344
• Forwards 9,593
• Swaps 2,444
• Options 2,307
• Equity-linked contr. 1,809
• Forw. and swaps 283
• Options 1,527
• Commodity contr. 548
• Others 11,408

IR contracts 11,669 Futures 7,914 Options 3,7
56 FX contracts 59 Futures 37 Options 22 Sto
ck-index contr. 1,793 Futures 334 Options 1,
459
World GDP in 99 30,000B All stocks and bonds
70,000
Liquidation value 2,800B
Source BIS
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Global Derivatives Markets 2001
OTC Instruments 111T
Exchange traded 23.5T

• IR contracts 77,513
• FRAs 7,737
• Swaps 58,897
• Options 10,879
• FX contracts 16,748
• Forwards 10,336
• Swaps 3,942
• Options 2,470
• Equity-linked contr. 1,881
• Forw. and swaps 320
• Options 1,561
• Commodity contr. 598
• Others 14,375

IR contracts 21,614 Futures 9,137 Options 12,
477 FX contracts 89 Futures 66 Options 23 St
ock-index contr. 1,838 Futures 295 Options 1
,543
Source BIS
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Chapter 22Credit Derivatives

• Following P. Jorion 2001
• Financial Risk Manager Handbook

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Credit Derivatives

• From 1996 to 2000 the market has grown from
• 40B
• to
• 810B
• Contracts that pass credit risk from one
  counterparty to another. Allow separation of
  credit from other exposures.

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Credit Derivatives

• Bond insurance
• Letter of credit
• Credit derivatives on organized exchanges
• TED spread Treasury-Eurodollar spread
• (Futures are driven by AA type rates).

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Types of Credit Derivatives

• Underlying credit (single or a group of
  entities)
  
• Exercise conditions (credit event, rating,
  spread)
  
• Payoff function (fixed, linear, non-linear)

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Types of Credit Derivatives

• November 1, 2000 reported by Risk
• Credit default swaps 45
• Synthetic securitization 26
• Asset swaps 12
• Credit-linked notes 9
• Basket default swaps 5
• Credit spread options 3

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Credit Default Swap

• A buyer (A) pays a premium (single or periodic
  payments) to a seller (B) but if a credit event
  occurs the seller (B) will compensate the buyer.

B - seller
A - buyer
Reference asset
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Example

• The protection buyer (A) enters a 1-year credit
  default swap on a notional of 100M worth of
  10-year bond issued by XYZ. Annual payment is 50
  bp.
• At the beginning of the year A pays 500,000 to
  the seller.
  
• Assume there is a default of XYZ bond by the end
  of the year. Now the bond is traded at 40 cents
  on dollar.
  
• The protection seller will compensate A by 60M.

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Types of Settlement

• Lump-sum fixed payment if a trigger event
  occurs
  
• Cash settlement payment strike market
  value
  
• Physical delivery you get the full price in
  exchange of the defaulted obligation.
  
• Basket of bonds, partial compensation, etc.
• Definition of default event follows ISDAs Master
  Netting Agreement

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Total Return Swap (TRS)

• Protection buyer (A) makes a series of payments
  linked to the total return on a reference asset.
  In exchange the protection seller makes a series
  of payments tied to a reference rate (Libor or
  Treasury plus a spread).

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Total Return Swap (TRS)
B - seller
A - buyer
Reference asset
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Example TRS

• Bank A made a 100M loan to company XYZ at a
  fixed rate of 10. The bank can hedge the
  exposure to XYZ by entering TRS with counterparty
  B. The bank promises to pay the interest on the
  loan plus the change in market value of the loan
  in exchange for LIBOR 50 bp.
• Assume that LIBOR9 and by the end of the year
  the value of the bond drops from 100 to 95M.
  
• The bank has to pay 10M-5M5M and will receive
  in exchange 90.5M9.5M

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Credit Spread Forward

• Payment (S-F)DurationNotional
• S actual spread
• F agreed upon spread
• Cash settlement
• May require credit line of collateral
• Payment formula in terms of prices
• Payment P(yF, T)-P(yS,T)Notional

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Credit Spread Option

• Put type
• Payment Max(S-K, 0)DurationNotional
• Call type
• Payment Max(K-S, 0)DurationNotional

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Example

• A credit spread option has a notional of 100M
  with a maturity of one year. The underlying
  security is a 8 10-year bond issued by
  corporation XYZ. The current spread is 150bp
  against 10-year Treasuries. The option is
  European type with a strike of 160bp.
• Assume that at expiration Treasury yield has
  moved from 6.5 to 6 and the credit spread
  widened to 180bp.
  
• The price of an 8 coupon 9-year semi-annual bond
  discounted at 61.87.8 is 101.276.
  
• The price of the same bond discounted at
  61.67.6 is 102.574.
  
• The payout is (102.574-101.276)/100100M
  1,297,237

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Credit Linked Notes (CLN)

• Combine a regular coupon-paying note with some
  credit risk feature.
  
• The goal is to increase the yield to the investor
  in exchange for taking some credit risk.

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CLN
A buys a CLN, B invests the money in a high-rated
investment and makes a short position in a credit
default swap. The investment yields LIBORYbp, th
e short position allows to increase the yield by
Xbp, thus the investor gets LIBORYX.
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Credit Linked Note
CLN AAA note Credit swap
Credit swap buyer
investor
AAA asset
Asset backed securities can be very dangerous!
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Types of Credit Linked Note

• Type Maximal Loss
• Asset-backed Initial investment
• Compound Credit Amount from the first default
• Principal Protection Interest
• Enhanced Asset Return Pre-determined

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FRM 1999-122 Credit Risk (22-4)

• A portfolio manager holds a default swap to hedge
  an AA corporate bond position. If the
  counterparty of the default swap is acquired by
  the bond issuer, then the default swap
• A. Increases in value
• B. Decreases in value
• C. Decreases in value only if the corporate bond
  is downgraded
  
• D. Is unchanged in value

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FRM 1999-122 Credit Risk (22-4)

• A portfolio manager holds a default swap to hedge
  an AA corporate bond position. If the
  counterparty of the default swap is acquired by
  the bond issuer, then the default swap
• A. Increases in value
• B. Decreases in value it is worthless (the same
  default)
  
• C. Decreases in value only if the corporate bond
  is downgraded
  
• D. Is unchanged in value

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FRM 2000-39 Credit Risk (22-5)

• A portfolio consists of one (long) 100M asset
  and a default protection contract on this asset.
  The probability of default over the next year is
  10 for the asset, 20 for the counterparty that
  wrote the default protection. The joint
  probability of default is 3. Estimate the
  expected loss on this portfolio due to credit
  defaults over the next year assuming 40 recovery
  rate on the asset and 0 recovery rate for the
  counterparty.
• A. 3.0M
• B. 2.2M
• C. 1.8M
• D. None of the above

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FRM 2000-39 Credit Risk

• A portfolio consists of one (long) 100M asset
  and a default protection contract on this asset.
  The probability of default over the next year is
  10 for the asset, 20 for the counterparty that
  wrote the default protection. The joint
  probability of default is 3. Estimate the
  expected loss on this portfolio due to credit
  defaults over the next year assuming 40 recovery
  rate on the asset and 0 recovery rate for the
  counterparty.
• A. 3.0M
• B. 2.2M
• C. 1.8M 1000.03(1 40) only joint default
  leads to a loss
  
• D. None of the above

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FRM 2000-62 Credit Risk (22-11)

• Bank made a 200M loan at 12. The bank wants to
  hedge the exposure by entering a TRS with a
  counterparty. The bank promises to pay the
  interest on the loan plus the change in market
  value in exchange for LIBOR40bp. If after one
  year the market value of the loan decreased by 3
  and LIBOR is 11 what is the net obligation of
  the bank?
• A. Net receipt of 4.8M
• B. Net payment of 4.8M
• C. Net receipt of 5.2M
• D. Net payment of 5.2M

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FRM 2000-62 Credit Risk (22-11)

• Bank made a 200M loan at 12. The bank wants to
  hedge the exposure by entering a TRS with a
  counterparty. The bank promises to pay the
  interest on the loan plus the change in market
  value in exchange for LIBOR40bp. If after one
  year the market value of the loan decreased by 3
  and LIBOR is 11 what is the net obligation of
  the bank?
• A. Net receipt of 4.8M (12-3)
  (110.4)200M
  
• B. Net payment of 4.8M
• C. Net receipt of 5.2M
• D. Net payment of 5.2M

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Pricing and Hedging Credit Derivatives

• 1. Actuarial approach historic default rates
• relies on actual, not risk-neutral probabilities
• 2. Bond credit spread
• 3. Equity prices Mertons model

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Example Credit Default Swap

• CDS on a 10M two-year agreement.
• A protection buyer agrees to pay to
• B protection seller a fixed annual fee in
  exchange for protection against default of 2-year
  bond XYZ.
  
• The payout will be Notional(100-B) where B is
  the price of the bond at expiration, if the
  credit event occurs.
  
• XYZ is now A rated with YTM6.6, while T-note
  trades at 6.

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Actuarial Method

• 1Y 1 probability of default
• 2Y 0.010.900.020.070.050.021.14

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Actuarial Method

• 1Y 1 probability of default
• 2Y 0.010.900.020.070.050.021.14
• If the recovery rate is 60, the expected costs
  are

1Y 1(100-60) 0.4 2Y 1.14(100-60) 0
.456
Annual cost (no discounting)
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Credit Spread Method

• Compare the yield of XYZ with the yield of
  default-free asset. The annual protection cost
  is
  
• Annual Cost 10M (6.60-6) 60,000

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Equity Price Method

• Following the Mertons model (see chapter 21) the
  fair value of the Put is

The annual protection fee will be the cost of Put
divided by the number of years.
To hedge the protection seller would go short the
following amount of stocks