4119Swaps

1
Derivative Securities

Swaps
2
Currency and Interest Rate Swaps

• A swap agreement between two parties commits each
  counterparty to exchange an amount of funds,
  determined by a formula, at regular intervals,
  until the swap expires.
• In the case of a currency swap, there is an
  initial exchange of currency and a reverse
  exchange at maturity.
• The exchange of two currencies at the current
  exchange rate with an agreement to reverse the
  trade -- at the same exchange rate -- at some set
  date in the future. One of the parties will pay
  the other annual interest payments.

3
Currency and Interest Rate Swaps

• A swap is equivalent to a collection of forward
  contracts that call for an exchange of funds at
  specified times in the future.
• However, it reduces transaction costs by allowing
  the counterparties to arrange in one transaction
  (the swap) what would take many transactions
  (using forward contracts) to replicate.
• In addition, the legal structure of a swap
  transaction may have advantages that reduce the
  risk to each party in the event of a default by
  the other party.

4
Origins and Underpinningsof the Swap Market

• In the early 1980s, the currency swap evolved as
  a way to simplify and speed the exchange of
  currency cash flows between counterparties, and
  quickly gained popularity.
• The use of a swap lowers the transaction costs.
• It links the two cash flows - only the net
  difference has to be paid, and there may be a
  right-of-offset.
• As a new financial product, it was also not
  covered by any accounting disclosure or security
  registration requirements.

5
Origins and Underpinningsof the Swap Market

• The notional value of outstanding swaps grew
  rapidly from zero in 1980 to more than 45
  trillion in 1999.
• The notional value is the underlying amount on
  which swap payments are based, while the gross
  market value is the cost that one party would
  have to pay to replace a swap at market prices in
  the event of a default.

6
Origins and Underpinningsof the Swap Market

• The Bank for International Settlements (BIS)
  estimated the gross market value in 1999 to be
  about 250 billion for currency swaps (10.2 of
  notional value), and 1,150 billion for interest
  rate swaps (2.6 of notional value).
• Of the 900 billion of daily volume that is not
  from the spot market, 734 billion of this is FX
  swaps (BIS 1999) the derivatives market have
  grown up around the spot market!
• The gross market value represents the gross
  exposure associated with swap contracts.
• However, the use of bilateral netting and
  collateral arrangements can significantly reduce
  the actual exposure.

7
Example

• Suppose we have two companies Air Canada (AC)
  and American Airlines (AA). AC would like to
  borrow US dollars (USD) and AA Canadian dollars
  (CAD).
• The borrowing rates are as follows
• Both companies can benefit if AC
• borrows CAD at 10 and AA
• borrows USD at 7 and then swap
• currencies.
• The gains of doing this are (12-10) (7-6)
  1
• If we assume that the benefit is equally split
  between the parties, through the swap, AC will
  face a lower 5.5 borrowing rate on USD and AA
  will pay 11.5 rate on CAD.

8
Swap of debt payments

• Each company issues fixed-rate debt in a currency
  that is available to it then the two companies
  swap the proceeds of the debt issue and also
  assume each others obligation to make interest
  and principal payment.
• IBM and World Bank 1981 Salomon Brothers as
  intermediary (pages 501- 505)
• IBM had DM and Swiss franc loans (had absolute
  advantage in the SFr bond market) in 1981
  (borrowed in earlier years), pay interest in DM
  and SF
• World Bank issued eurodollar bonds (had absolute
  advantage in the US bond market)
• IBM and World Bank swap debt payments IBM pay
  for WB, WB pay DM and SF for IBM

9
The IBM-World Bank Swap
10
The IBM-World Bank Swap - details

• IBM owed 12.375 M SFr in interest and 200 M SFr
  (principal) 30 M DM (interest) and 300 M DM
  (principal) - and wanted the WB to pay for it.
• How much would IBM pay in US dollars for the WBs
  debt?
• Intr (Swiss)8, Intr (Germany)11

11
The IBM-World Bank Swap - details

• How much would IBM pay in US dollars for the WBs
  debt?
• Answer PV(DM)PV(SFr)205.485 M USD
• The WB could borrow this amount and have a
  payment schedule
• matching the schedule of the IBM debt.
• IBM would be making payments in USD for the WB

12
The Basic Cash Flows of a Currency Swap

• Firms A and B can each issue a 7-year bond in
  either the US or SFr market. Assume firm A wants
  SFr and firm B wants US.
• Firm A enjoys an absolute advantage in both
  credit markets.

13
The Basic Cash Flows of a Currency Swap

• Firm A has a comparative advantage in borrowing
  US, while firm B has a comparative advantage in
  borrowing SFr.

• By borrowing in their comparative advantage
  currencies and then swapping, lower cost
  financing is possible (borrowing at a lower cost).

14
The Basic Cash Flows of a Currency Swap

• Together, A (-10-5.510.75-4.75 on SFr loan)
  and B (-6-10.755.5-11.25 on US loan) save
  0.5. Note that if a bank or swap dealer
  intermediates the transaction and charges a fee,
  the aggregate interest savings will be reduced.

15
Currency Swap

• Another example.
• A multinational company has just received 1
  million from sales and knows it will have to pay
  those dollars to a BC supplier in three months.
• Meanwhile, the company would like to invest in
  the 1 million in British pounds.
• A three-month swap of dollars into British pounds
  may result in lower brokers fees than the two
  separate transactions of selling for spot British
  pounds and selling the British pounds for dollars
  on the forward market.

16
The Basic Cash Flows of a Currency Swap

• What is the source of these financing savings?
  Are they consistent with an efficient
  international financial market?
• Capital market segmentation. This is a case where
  two capital markets reach different evaluations
  about two firms and apply a different interest
  cost conditional on the risk of the firm.
• Saturation and scarcity value. An issuer who has
  not saturated the market may enjoy a scarcity
  value and be able to issue bonds at a lower rate.

17
The Basic Cash Flows of a Interest Rate Swap

• Firms A and B can each issue a 7-year US
  denominated bond in either fixed-rate or
  floating-rate terms. Assume firm A wants a
  floating rate and firm B wants a fixed rate loan.
• Firm A enjoys an absolute advantage in both
  credit markets.

18
The Basic Cash Flows of a Interest Rate Swap

• Firm A has a comparative advantage in the
  fixed-rate bond market, while firm B has a
  comparative

advantage in the floating-rate bond market.

• By borrowing in their comparative advantage
  markets and then swapping, lower cost financing
  is possible.

19
The Basic Cash Flows of a Interest Rate Swap

• Together, A and B save 1. Note that if a bank or
  swap dealer intermediates the transaction and
  charges a fee, the aggregate interest savings
  will be reduced.

20
(Fixed rate) Currency Swap

• Equivalent to buying a fixed rate bond in
  currency A (B) and selling a fixed rate bond in
  currency B (A) and exchanging notional values at
  maturity.
• Suppose we want to have currency B we borrow A
  (sell a bond), lend it to the counter-party pay
  interest ( notional value) payments in currency
  B.
• Suppose B is the foreign currency we would be
  paying interest on B and receive interest on A.
• At maturity prices of bonds PB(T) and PBF(T)
• PB(T)St x PBF(T)

21
Currency Swap

• Question 6 from the text (pp. 531)
• Swap requires payments at dates 2, 3, 4
• A U.S (home) coupon bond pays a coupon (8) at
  dates 2, 3, 4 and is selling at par
• A foreign coupon (5) bond is selling at par
• S0.5/foreign currency
• Notional amount of the swap is 50 M
• What is the foreign currency fixed rate payment
  (made by us)?
• continued -gt

22
Currency Swap

• Since both bonds are selling at par, the term
  structures in both countries are flat at the
  coupon rate.
• The notional amount in foreign currency is
  50M/0.5 100 M
• Thus, the foreign fixed rate payment, c, can be
  obtained from
• 100M c/1.05 c/1.052 (100c)/1.053
• This yields c5M

23
Currency Swap

• In general, formula for the pricing of a
  default-free fixed or floating swap with N
  payments and notional amount M

24
Currency Swap

• Question 7 from the text (pp. 531)
• At date 2, rUS8 (unchanged)
• The foreign term structure has shifted up by
  100bp1
• S20.6/foreign currency
• What is the value of the swap for the U.S fixed
  rate payer?
• continued -gt

25
Currency Swap

• Value of the US bond is still at par 50M.
• The value of the foreign bond is
• 0.6 x (5M/1.06 105M/1.062) 58.89M
• Therefore, the value of the swap is 58.89M -
  50M 8.89 M.

26
Currency Swap

• Question 8 from the text (pp. 531)
• S40.8/foreign currency
• What is the net payment that the U.S. fixed rate
  payer receives?
• Answer
• At date 4, the value of US bond is 54M (notional
  valuelast coupon).
• At date 4, the dollar value of foreign bond is
  105M x 0.8 84M.
• Thus, the net payment that the US fixed-rate
  payer receives is 30M.

27
Interest Rate Swap

• We will focus on a fixed/floating rate swap
• A swap must be priced so that the present value
  of the fixed rate payments equals the present
  value of the floating rate payments
• The swap interest payments are computed based on
  a notional amount that is never exchanged.
• This is not the case with currency swaps.

28
Interest Rate Swap

• Question 1 from the text (pp. 531)
• An interest rate swap with notional amount of
  100 M is initiated at date 0.
• The bond matures at date 4.
• LIBOR for one period is 5.
• Interest rates for bonds maturing at dates 1, 2,
  3, 4 are 5, 5.5, 5.5, 5.75, respectively.
• Payments take place at dates 2, 3, 4.
• Find interest payment c that has to be made at
  dates 2, 3, 4 for the swap to have no value at
  initiation.

29
Interest Rate Swap

• Answer
• The value of floating rate bond immediately
  before the start of an interest payment period is
  always equal to par.
• Thus, at date 1, the value of the floating rate
  bond is
• (100M LIBOR x 100M)/(1LIBOR) 100M.
• The value of fixed rate bond at date 1 is
• c/1.055 c/1.0552 (100Mc)/1.05753100 M
• gtc 5.73615M

30
Interest Rate Swap

• Question 2 from the text (pp. 531)
• The swap from Q1 is modified so that LIBOR never
  pays more than 6 per period.
• This is swap with option (making sure that if you
  are to pay LIBOR, you are insured against
  increases in LIBOR) .
• The price of a cap with exercise price of 6 on
  100 M notional is 1 M (premium).
• Find new interest payment c that has to be made
  at dates 2, 3, 4 for the swap to have no value at
  initiation.

31
Interest Rate Swap

• Answer
• Now, the amount that will be received by the
  fixed rate payer is reduced by the value of the
  cap.
• Thus, c can be obtained from
• 100M - 1M
• c/1.055 c/1.0552 (100 Mc)/1.05753
• gtc5.36467 M
• Note that c lt c!

32
Interest Rate Swap

• Question 3 from the text (pp. 531)
• At date 2, the term structure of zero-coupon
  bonds is flat at 6.
• LIBOR for one period is 6.
• What is the value of the swap from Q1 for the
  fixed-rate payer?
• Answer
• The fixed rate payments and floating rate
  payments are compared at date 2 right after the
  payment is made
• ...continued-gt

33
Interest Rate Swap

• The fixed rate payer will pay 5.73615M at date 3
  and 105.73615M at date 4.
• The present value of the payment as of date 2 is
  5.73615M/1.06 105.73615M/1.062 99.516259M.
  
• The present value of what the fixed rate payer
  will receive is at par since it is right before
  the new payment period starts.
• Thus, the value of the swap is 100M -
  99.516259M 483,741.

34
Interest Rate Swap

• Question 4 from the text (pp. 531)
• All numbers from Q1.
• But, now, payments are made ad dates 3 and 5.
• The interest rate per period for a zero-coupon
  bond maturing at date 5 is 6.
• Compute c that has to be made at dates 3 and 5.

35
Interest Rate Swap

• Answer
• We assume that only fixed-rate payments are made
  at dates 3 and 5.
• The floating-rate payments are made at each date.
• Now the value of the fixed-rate bond at date 1 is
  c/1.0552 (100M c)/1.064 100 M
• gt c 12.2982M

36
Interest Rate Swap

• Question 5 from the text (pp. 531)
• At date 1, the term structure is flat at 6.
• What is the value of the swap from Q4 for the
  floating rate payer at date 2 (typo)?
• Answer
• Value of fixed rate payments at date 2 if the
  term structure is flat at 6 is 12.2982M/1.06
  112.2982M/1.063 105.89M.
• ...continued-gt

37
Interest Rate Swap

• Value of floating rate payment is always at par.
• Thus, the value of the swap is 105.89M 100M
  5.89M.

38
Interest Rate Swap

• Question 10 from the text (pp. 531)
• All as in Q1.
• However, now, the floating rate payer receives 2
  times the fixed rate and pays 2 times the
  floating rate.
• LIBOR6
• The term structure is flat at 6.
• Compute the value of the swap at dates 1 and 2.

39
Interest Rate Swap

• Answer
• The floating-rate leg of the swap is equivalent
  to a position in a floating rate bond with
  principal amount of 200M.
• The fixed-rate leg of the swap has payments
  corresponding to the payments a fixed-rate bond
  with principal amount of 200M.
• Everything in Q1 is multiplied by 2.
• c6 of 100 M 0.06 M