Objectives:

1
Hedging Interest Rate Risk

• Objectives
• Analyze the value of interest rate swaps and how
  they effect the duration of a FIs net worth.
• Analyze the value of forward rate agreements and
  how they effect the duration of a FIs net worth.
• Value forward and futures contracts on bonds and
  determine how they effect the duration of a FIs
  net worth.
• Consider how interest rate options such as bond
  options, caps, floors, collars, and swaptions can
  insure against interest rate risk.

2

• We have analyzed how to value fixed-cashflow and
  floating-cashflow bonds and loans.
• We also have discussed how to determine the
  duration (interest rate risk) of these fixed and
  floating rate instruments as well as the duration
  of a financial intermediarys (FIs) net worth.
• The best way to limit the interest rate risk of a
  FIs net worth may not be to match the durations
  of its assets and liabilities.
• It may be best to let the FIs customers
  determine the durations of its assets and
  liabilities, and then to control the resulting
  interest rate risk using interest rate
  derivatives.

3
Interest Rate Swaps

• Swap contracts are agreements between two parties
  to exchange a series of assets or cash flows at
  specified future dates.
• In particular, an interest rate swap is a
  contract to exchange interest payments on a given
  amount of notional principal at specified
  future dates.
• The most common type of (plain vanilla) interest
  rate swap is where one party pays floating rate
  interest payments (usually equal to six-month
  LIBOR) in exchange for the other party paying
  fixed rate interest payments.
• These swaps are sold by dealers in large banks to
  FIs and other corporations that seek to hedge
  interest rate risk.

4

• Example A banks depositors usually prefer short
  maturities that can be readily liquidated. Its
  borrowing customers may prefer longer maturity
  loans at fixed-interest rates because the loans
  finance longer maturity projects and borrowers
  wish to know their exact fixed-interest borrowing
  costs.
• If a bank satisfies both its borrowing and
  depositor customers, its longer duration loans
  financed by shorter duration deposits exposes the
  bank to interest rate risk.
• Consider a bank that makes a 1 million,
  five-year loan at a fixed 8 interest rate,
  where the borrower is required to make
  semi-annual payments of ½(.08)(1 m) 40,000
  and repay the 1 m principal at maturity. The
  bank finances this loan by issuing 1 m of
  six-month maturity CDs.

5

• These transactions expose the bank to interest
  rate risk. If
• six-month market interest rates rise above 8
  , the banks
• interest payments on its CDs will exceed its
  8 loan payments.

Unhedged Bank Balance Sheet
Assets Liabilities Fixed-Rate 5
year Loan Six-Month CDs (Long
Duration) (Short Duration)

• The bank can hedge this interest rate exposure
  by becoming a
• fixed-rate payer in a five-year interest rate
  swap contract. The
• swap agreement will require the bank to pay
  semi-annual
• interest at a 7 annual rate on a notional
  principal of 1 m.
• In return, the bank receives semi-annual
  six-month LIBOR
• payments. With this swap, the banks on and
  off-balance sheet
• assets and liabilities are now

6
Bank Balance Sheet
Assets
Liabilities Fixed-Rate 5 year Loan
Six-Month CDs (Long Duration)
(Short Duration)
Off-Balance Sheet (Swap) Assets
Liabilities LIBOR Swap Payments
Fixed-Rate Swap Payments (Short
Duration) (Long Duration)

• Because six-month LIBOR will move with the
  interest rate that
• the bank pays on its CDs, the LIBOR swap
  payments will
• cover the banks required CD interest
  payments. The net result
• is that the duration of on- and off- balance
  sheet assets are
• equal to the duration of on-and off-balance
  sheet liabilities.

7

• How can we value an interest rate swap and
  calculate its effect on the duration of a FIs
  net worth? This is easy once we recognize that
  an interest rate swap is identical to opposite
  positions in a fixed-coupon bond and a floating
  coupon bond.
• A floating (fixed) rate payer in an n-year
  interest rate swap has a long (short) position in
  an n-year fixed-coupon bond and a short (long)
  position in an n-year floating-coupon bond.
• Example An insurance company has longer duration
  liabilities (insurance policies) than assets. To
  hedge its risk, it becomes a floating rate payer
  (fixed rate receiver) in a 5 m. 10-year swap
  tied to 6-month LIBOR and having a fixed-rate of
  6 .
• Thus, the value of this swap is the difference
  between a 5 m., 10-year coupon bond with an
  annual coupon rate of 6 less a 5m., 10-year
  floating rate bond tied to six-month LIBOR.

8

• Specifically, consider a swap having exactly n
  years until maturity and making semi-annual
  payments, with the next payment being exactly 6
  months from now. Let
• FV swaps notional principal.
• c annual coupon rate of swap (½cFV paid every 6
  months).
• i½(?) be the semi-annually compounded yield on a
  ZCB that matures in ? years, so that P(?)
  1/1½i½(?)2?.
• The value of the fixed-rate component of the
  swap, PVfx, is simply the present value of an
  n-year fixed-coupon bond

9

• The value of the floating rate component of the
  swap, PVfl , is
• PVfl FV
• because a floating-rate bond equals its par
  value at the coupon reset date.
• Hence, the value of the swap to the floating rate
  payer (fixed rate receiver) is
• When the two parties first agree to the swap, its
  value equals zero because the market-determined
  coupon rate, c, is set so that the fixed-rate
  component of the swap sells for its par value,
  FV, that is, c equals the fixed-rate bonds
  initial YTM.

10

• However, after the swap is agreed to, changes in
  market interest rates imply that PVswap will have
  a negative (positive) value if market rates
  increase (decrease).
• Example A 5-year 100 m. swap was agreed to 3
  years ago with a c 8.00 coupon rate.
  Currently, the swap has exactly 2 years to
  maturity and the semi-annually compounded yields
  on ½, 1, 1½, and 2 year ZCBs are 3.0, 3.2,
  3.4, and 3.6, respectively. What is the
  current value of the swap?
• Therefore

11

• It is also straightforward to calculate the
  impact of a swap on the duration of a FIs net
  worth. As before, let
• A value of FIs on-balance sheet assets
• DA duration of FIs on-balance sheet assets
• L value of FIs on-balance sheet liabilities
• DL duration of FIs on-balance sheet
  liabilities
• PVfx value of fixed component of swap
• Dfx duration of fixed component of swap
• PVfl value of floating component of swap
• Dfl duration of floating component of swap
• DE duration of FI net worth
• Note that when the next swap payment is exactly 6
  months in the future, PVfl FV and Dfl ½ year.
  Also, when the parties initially agree to the
  swap, then PVfx FV , but can change afterwards
  as the previous example shows.

12

• We know how to calculate the duration of the
  fixed component, since it is simply the duration
  of a fixed-coupon bond
• For the previous example, this equals

13

• Now consider the case (as in the previous bank
  example) where the FI is a fixed-rate payer in an
  interest rate swap with notional principal of FV
  and the next swap payment is 6 months in the
  future. Then the FIs net worth, E, equals
• and the duration of net worth is
• When the swap is initially agreed, PVfx FV and
  then

14

• Example A bank has assets of 1.1 billion and
  debt (deposits) of 1.0 billion. The durations
  of its assets and debt are 1.25 years and 1 year,
  respectively. It now agrees to be a fixed-rate
  payer (floating rate receiver) in a 5-year
  interest rate swap having a notional principal of
  80 m. The duration of the fixed-component of the
  swap is 4.5 years. What is the the duration of
  the banks net worth?

15
Forward Rate Agreements

• A forward rate agreement (FRA) is simply an
  exchange of fixed and floating interest payments
  for a single future date. Thus, it is like an
  interest rate swap covering a single period.
• As before, assume that two parties swap floating
  6-month LIBOR interest for a fixed interest rate
  of c based on a notional principal of FV. This
  swap is assumed to occur ? gt ½ years in the
  future.
• As with swaps and floating rate bonds, the
  6-month LIBOR rate for the FRA is determined ? -
  ½ years in the future.
• From our previous analysis of floating rate
  coupons, the floating rate cashflow is replicated
  by buying a ZCB maturing in ? - ½ years and
  issuing a ZCB maturing in ? years.

16

• Hence, the present value of the floating rate
  payment is
• The present value of the fixed-coupon payment of
  ½cFV is
• Thus, the value of a FRA to receive floating and
  pay fixed is
• and the effect on a FIs net worth is to
  decrease its duration by adding off-balance sheet
  assets worth FV P(? - ½ ) and having a duration
  of ? - ½ years and adding off-balance sheet
  liabilities worth FV(1½c)P(? ) and having a
  duration of ? years.

17

• When the FRA is initiated, its value equals zero.
  This implies that the present values of the
  added off-balance sheet assets and liabilities
  are equal. This insight also determines c
• c is then the semi-annually compounded forward
  rate for an investment starting ? - ½ years and
  ending ? years in the future.
• Note that a FRA to pay fixed (floating) and
  receive floating (fixed) would be the appropriate
  hedge for a FI that wishes to convert a future
  on-balance sheet fixed-(floating) rate loan or
  bond interest cashflow (asset) to a floating
  (fixed) one.

18
Forward and Futures Contracts on Bonds

• A forward contract is an agreement between two
  parties to exchange a particular asset, such as
  a bond, at a specified future date at a pre-
  agreed price.
• The party taking the long position agrees to
  purchase this asset from the party taking the
  short position who agrees to deliver the asset.
• The price that is paid by the long party to the
  short party at the future date is called the
  forward price. The parties agree to this price
  when the contract is first established.
• Let a forward contract have a time until maturity
  of ? years and let f be the forward price paid by
  the long party to the short party in return for a
  particular bond.

19
Long party pays f Short party delivers bond
t? Contract Maturity Date
t Current date

• If the bond does not pay any cashflows (coupons)
  during the life of the forward contract (from
  dates t to t?), the value of this forward
  contract to the long party is simply the present
  value of the bond minus the present value of the
  cashflow f to be paid at date t?.
• Moreover, the effect of this contract on the
  duration of a FIs net worth is to add
  off-balance sheet assets worth PVbond with a
  duration equal to that of the bond and adding
  off-balance sheet liabilities worth fP(? ) with a
  duration of ? years.

20

• Example A forward contract on the 10-year U.S.
  Treasury note matures in exactly 3 months and has
  a forward price of f 101. The Treasury notes
  current price is 103 and its duration is 9
  years. The annualized, quarterly-compounded
  3-month LIBOR is 4.00 . What is the value of
  this forward contract to the long party?
• The contract would be equivalent to adding 103
  of assets with a duration of 9 years and adding
  100 of liabilities with a duration of 3 months.
  Hence a long (short) forward position would
  lengthen (shorten) the duration of a FI net
  worth.

21

• When a forward contract is first initiated, the
  forward price, f, is set so that the contracts
  value equals zero. This implies that at the
  initial date
• If bond pays cashflows (coupons) during the life
  of the forward contract (from dates t to t?),
  let PVc be their present value. Because the long
  party does not receive them, the value of the
  forward contract is adjusted to be

22

• Like swaps and FRAs, most forward contracts are
  bought and sold in over-the-counter markets where
  large banks act as market makers (dealers).
• Futures contracts are like forward contracts in
  that they involve agreements to exchange assets
  at future dates. Their profitability and uses in
  hedging are very similar to forward contracts.
  However futures contracts are re-traded through
  time at a centralized exchange, such as the CBOT,
  CME, or LIFFE.
• The U.S. Treasury futures traded on the CBOT is
  very similar to the just-described U.S. Treasury
  bond forward contract.
• Eurodollar futures traded on the CME is very
  similar to the previously described FRAs but are
  tied to 3-month LIBOR, rather than 6-month LIBOR.
  (The valuation and duration formulas are similar
  but change ½ to ¼).

23
Interest Rate Options

• Unlike forwards, futures, and swap contracts
  where parties are committed to carrying out
  future transactions, the owner of an option has
  the right, but not the obligation, to execute a
  future transaction.
• The owner of a call option written on an asset
  has the right, but not the obligation to buy an
  asset at some future date, T, for the pre-agreed
  exercise or strike price, X. Thus, if the
  assets price at date T is ST, a call options
  payoff at maturity is
• Call option payoff max
  ST - X, 0
• Similarly, the owner of a put option written on
  an asset has the right, but not the obligation to
  sell an asset at some future date, T, for the
  pre-agreed exercise price, X. Its payoff at
  maturity is
• Put option payoff max X
  ST , 0

24

• If, for example, X 100, a call options
  payoff can be
• graphed as

Payoff at Maturity of Call
Option Payoff 100
0

-100
0 X100
200 Date T asset
price, ST
25

• For example, if X 100, a graph of the put
  options payoff
• will be

Payoff at Maturity of Put
Option Payoff 100
0

-100
0 X100
200 Date T asset
price, ST
26

• Options can never have a payoff to the owner of
  less than zero, since the owner can always
  choose to not exercise them. Hence, the owner
  must pay the seller (or writer) of the option an
  initial premium for the right.
• Note that a call (put) option is a bullish
  (bearish) position in the underlying asset. Its
  value increases (decreases) as the value of the
  underlying asset increases.
• If the underlying asset is a bond (e.g., 10-year
  U.S. Treasury note), then a call options value
  goes up (down) when market interest rates fall
  (rise). Conversely, a put options value goes
  down (up) when market interest rates fall (rise).
• Valuing an interest rate option and computing its
  duration can be done using extensions of the
  Black-Scholes model, topics that are beyond the
  scope of this course.

27

• However, we can illustrate how interest rate
  options can be used to insure against risks.
• Example An insurance company holds a portfolio
  of bonds that have a duration of 8 years. The
  current value of its portfolio equals S0 10
  million. It wishes to insure itself against a
  fall in the value of its portfolio below X 9 m
  during the next year.
• This could be accomplished by purchasing put
  options on a bond that has a duration of
  approximately 9 years (9 years minus the option
  contract maturity of one year).
• The insurance company purchases put options on
  the 10-year U.S. Treasury note having a current
  underlying bond value of S0 10 m, an effective
  exercise price of X 9 m, and a maturity of one
  year.

28

• If we let I1 be the value of the put options
  (insurance) at the end of the year, then their
  value will be
• where S1 is the end-of-year value of the
  Treasury notes.
• By owning this portfolio insurance, the
  insurance company will have a combined
  end-of-year value given by
• so that it is guaranteed to be worth no less
  than 9m. This end-of-year value for the
  insurance companys assets can be graphed as

29
End-of-Year Insurance Company
Assets Value 13m
11m 9m 7m


5m 5m 7m 9m
11m 13m Value
of Bonds
30

• Another option contract that provides insurance
  is an interest
• rate cap. Consider a firm that has borrowed by
  issuing a floating
• rate bond paying semi-annual coupons tied to
  six-month LIBOR.
• The firm can obtain insurance from paying very
  high interest
• rates by purchasing interest rate caps from its
  bank.

• A cap is an agreement where the bank will pay
  the borrower
• the difference between six-month LIBOR, i½,T ,
  and an agreed
• upon cap rate, RC , whenever LIBOR exceeds the
  cap rate

Maturity Value of Cap ½ FV max 0, i½,T - RC

where FV is the notional principal underlying the
cap agreement. The bank makes this payment six
months after the above LIBOR, i½,T, is set.
Purchasing caps for every date that the floating
rate bond makes semi-annual coupon payments
insures that the borrower never pays more than RC.
31

• Conversely, an interest rate floor is a contract
  where the seller agrees to pay the buyer the
  difference between a floor rate, RF , and LIBOR
  when LIBOR is below the floor rate
• An interest rate collar is the purchase of an
  interest rate cap and the sale of an interest
  rate floor where RC gt RF. A floating rate note
  issuer that obtains a collar insures that its net
  interest payment will always be in the range of
  RC , RF.
• Another interest rate option is a swaption the
  right, but not the obligation, to initiate an
  interest rate swap agreement at a future contract
  date and at a pre-agreed swap rate, c.

Maturity Value of Floor ½ FV max 0, RF - i½,T

32

• Example The buyer of a swaption has the right,
  but not the obligation, to become the fixed-rate
  payer (floating rate receiver) in a 10-year
  interest rate swap beginning T 1 year from now
  and at a fixed-coupon rate of c 6.
• If, after one year, the market fixed-coupon rate
  on a 10-year swap is greater (less) than 6 ,
  this swaption would expire in-the-money
  (worthless).
• By buying this swaption, the holder would be
  insured against paying a fixed rate exceeding 6.
  This swaption would be valuable when market
  interest rates unexpectedly increase.
• Conversely, a swaption in which the buyer has the
  right to be the floating rate payer (fixed rate
  receiver) would be valuable when market interest
  rates unexpectedly decline.