EXAMPLE OF CROSS EXCHANGE RATES

1
EXAMPLE OF CROSS EXCHANGE RATES

• Consider the following example of calculating the
  range of cross bid and ask rates
• Suppose that
• ? DM/ask DM1.7295/, DM/bid DM1.6982/,
  /bid 0.753/, and /bid 1.283/
• What is the lowest bid price and the highest ask
  price for that a bank will offer in exchange
  for DM?

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• For a bank to attract business, it must offer a
  rate /bidDM such that
• /bidDM ? /bidDM ? /bid
• Using the fact that i/ask(j) 1/(j/bid(i)), we
  rewrite as
• DM/ask ? DM/ask ? /ask
• This is highest ask price for

3

• We know that /ask 1/(/bid) 1/0.753
  1.328/
• Thus
• DM/ask ? DM1.7295/ ? 1.328/
• or
• DM/ask ? DM2.296/

4

• Similarly, for a bank to attract business, it
  must offer a rate DM/bid such that
• DM/bid ? /bid ? DM/bid
• or
• DM/bid ? 1.283/ ? DM1.6982/
• or
• DM/bid ? DM2.178/
• This is the lowest bid price for

5
INTEREST PARITY CONDITIONS

• Consider a Belgian investor faced with the
  following options
• ? Invest BF100 in a 3-month domestic (BF
  denominated) security
• ? Invest BF100 in a 3-month foreign (
  denominated) security

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• After 3 months, the investor will receive
• ? BF 100 (1.25rBF) from investing in the
  domestic security (rBF is the annual nominal
  interest rate in BF)
• ? BF 100 (Se/S) (1.25r) from investing
  abroad assuming that the Belgian investor wants
  to hold BF only
• In the second case, the investor converts the BF
  into , invests in the security, and convert
  the back into BF after 3 months

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• The last calculation involves uncertainty about
  the future spot BF/ rate
• The investor needs to form an expectation of the
  future spot rate (Se)
• For the investor to be indifferent between the
  two options, it must be that
• BF 100 (1.25rBF) BF 100 (Se/S) (1.25r)

8

• Uncovered Interest Parity Condition (UIP) If
  investors treat investments in different
  currencies as perfect substitutes, then they will
  act to eliminate arbitrage profits
• The UIP condition is an unhedged interest parity
  condition because investors do not hedge for
  exchange-rate risk (assuming given interest
  rates)

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• Alternatively, an investor could hedge for
  exchange-rate risk through the forward market
• The forward exchange rate is the rate that is
  contracted today for the exchange of currencies
  at a specified date in the future
• The Belgian investor will purchase a forward
  contract to sell forward the amount of received
  after 3 months from investing in the security

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• We now can define the Covered Interest Parity
  Condition (CIP) as
• BF 100 (1.25rBF) BF 100 (F3/S) (1.25r)
• where F3 is the 3-month forward BF/ rate
• In the absence of transaction costs, any
  deviation from CIP implies that there is a
  risk-free arbitrage profit
• CIP must hold for currencies with forward
  exchange rates

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COVERED INTEREST ARBITRAGE

• If CIP does not hold, how should an investor
  proceed to capture the potential arbitrage
  profits?
• Two cases
• ? If (1r) (F/S) (1r), then borrow in
  foreign currency and invest in domestic currency

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• ? If (1r) domestic currency and invest in foreign currency
• With zero transaction costs, the forward rate is
  obtained from CIP (assuming given interest rates)
• If there exist transaction costs, then we must
  derive the range of forward bid and ask rates, as
  well as the steps of covered interest arbitrage

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COVERED INTEREST ARBITRAGE WITH TRANSACTION COSTS

• Define
• ? Sb bid rate of foreign currency in terms of
  domestic currency (e.g. C/bidFF shows how many
  C we get from selling one FF)
• ? Sa ask rate of foreign currency in terms of
  domestic currency (e.g. C/askFF)

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• ? Fb and Fa are the corresponding forwars rates
• An investor has two options
• ? Borrow domestically and invest abroad
• ? Borrow abroad and invest domestically
• E.g. Borrowing C100, a Canadian investor must
  repay after 3 months
• 100 (1.25rc)

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• Investing the C100 in France and covering for
  exchange-rate risk through the forward market,
  the investor receives after 3 months
• (100/Sa) (1.25rF) Fb
• So that there are no arbitrage profits, it must
  be that
• 100 (1.25rc) ? (100/Sa) (1.25rF) Fb
• or
• Fb ? Sa (1.25rc)/(1.25rF)

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• If the forward bid rate is greater than the
  right-hand-side quantity, then it is profitable
  to borrow domestically and invest abroad
• Alternatively, if the Canadian investor borrows
  in FF and invests domestically, then there exist
  no arbitrage profits if
• Fa ? Sb (1.25rc)/(1.25rF)

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• Both conditions must hold simultaneously so that
  there are zero arbitrage profits
• These conditions give the range of forward bid
  and ask rates when there exist transaction costs

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EXAMPLE OF COVERED INTEREST ARBITRAGE

• Suppose that
• ? r 5.96 per annum
• ? r 8.00 per annum
• ? S 1.5/
• ? F3 1.4925/
• Does CIP hold if we begin with 100?

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• Verify that
• 100 ? (1 0.0596 ? 0.25)
• 100 ? (1 0.08 ? 0.25) ? 1.4925/1.5
• What if F3 changes to 1.52/?
• Now, investing in is more profitable
• An investor must proceed as follows to capture
  the arbitrage profits

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• ? Step 1 Borrow 100 at 5.96
• ? Step 2 Purchase spot at 1.5/
• ? Step 3 Purchase securities at 8.00
• ? Step 4 Sell forward at 1.52/
• The profit from covered interest arbitrage will
  be 1.87 per 100

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• What if F3 changes to 1.48/?
• Now, an investor must proceed as follows
• ? Step 1 Borrow 100 at 8.00
• ? Step 2 Sell spot at 1.5/
• ? Step 3 Purchase securities at 5.96
• ? Step 4 Purchase forward at 1.48/

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• Borrowing 100, the investor will repay
• 100 ? (1 0.08 ? 0.25) 102
• Investing abroad, the investor will receive
• 100 ? (1.5/1.48) ? (1 0.0596 ? 0.25) 102.86
• The investor makes a profit of 0.86 per 100