1

International Economics

Foreign Exchange Calculations and Arbitrage

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Spot FX Market

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Structure of FX Market

RetailCustomers

InterbankCustomers

Speculators

CommercialBanks

F/X Brokers

CentralBank

CommercialBanks

CentralBank

Domestic Country Foreign Country

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Spot FX Market Transactions Commercial bank acts as broker for customers with foreign

exchange to buy or sell. As a broker, the bank quotes two types of rates for foreign currency.

Rates are normally quoted in terms of the rate at which the bank will buy or sell US dollars, primarily because the US dollar effectively functions as a "world" currency.– ASK Price, PASK Price, PAA: rate at which the bank will sell foreign currency in

return for domestic currency, i.e. USD.40/DM (or 2.50 DM/USD)– BID or OFFER Price, PBID or OFFER Price, PBB: rate at which the bank will buy foreign

currency in return for domestic currency, i.e. USD.38/DM (or 2.63 DM/USD)

– SPREADSPREAD: difference between the Bid and Ask price and is the way that the commercial bank makes a profit from buying and selling foreign exchange. Spread is usually quoted as a percent:

SPREAD = [(PSPREAD = [(PAA - P - PBB )/P )/PAA] ] xx 100 100

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Bid-Ask Spread in FX Markets Spread measures the cost of "in and out" of foreign exchange

so one-half the spread is the "cost" of a single transaction.– Exchange Rates reported in the paper are generally the Midpoint

Exchange Rate, defined as:

MIDPOINT EXR = [PMIDPOINT EXR = [PAA + P + PBB]/2]/2

Size of the spread that any given customer faces depends on several factors including;– the amount of foreign currency being traded (volume)

– the amount of competition faced by the bank

– the variability of the exchange rate over time (risk)

– the size of the market in the foreign currency (liquidity)

– the type of financial instrument being traded

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Mechanics of a Spot Transaction Spot FX transactions generally do not involve the actual

physical exchange of currencies across national borders.– Exception is individuals buying/selling foreign notes for travel.

Example: US Firm agrees to purchase equipment from a German Firm with the

price of the equipment denoted in DeutscheMarks (DM), say DM 100,000. US Firm will contact its US Bank and send the number of US dollars it will cost to buy 100,000 DM’s.

US Bank debits US Firm’s account for USD 50,000 (assume EXR of $0.50/DM) and credits German firm’s bank (German Bank) for the same USD 50,000. At the same time, German Bank will be notified of the transaction and will credit German Firm’s account with DM100,000 and debit the US Bank’s account for DM 100,000.

– These accounts between commercial banks are called Nostro and Vostro accounts, and are held by both banks to support the foreign exchange market.

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Mechanics of the Transaction

No currency has been transferred across national borders in this transaction. Instead, both the US bank and the German Bank have changed their holdings of US $ versus DM. – German Bank has more US $ than previously, while US Bank has fewer

DM than previously. This exposes each bank to risk if the USD/DM exchange rate changes in particular ways.

US Firm

US Bank

German Firm

German Bank

United States Germany

DepositsUS Firm -$X

German Bank +$X

DepositsGerman Firm +DM Y US Bank -DM Y

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Currency Cross-Rates Most currencies are quoted against the US$, so to get

exchange rate between two currencies need to be able to calculate currency cross rates given two spot or forward foreign exchange quotations involving three currencies;

• Set which currency is home currency & which is foreign currency. • Put exchange rates in terms of the third currency in denominator.• Divide one exchange rate by the other to eliminate the third currency.

Example Direct Cross Rate for Won in JapanExample Direct Cross Rate for Won in Japan:• Japanese Yen-US$ Direct Rate: ¥105.25/US$ • Korean Won-US$ Direct Rate: W1146.25/US$• What is the Direct Cross Rate for the Won in Japan?

Direct Cross Rate for Won in Japan = Direct Cross Rate for Won in Japan = ¥105.25/US$¥105.25/US$ W1146.25/US$W1146.25/US$

   Direct Cross Rate for Won in Japan = ¥0.092/WonDirect Cross Rate for Won in Japan = ¥0.092/Won More complicated examples of cross-rates take into account the

appropriate bid or ask rates.

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Currency Cross Rates

M o s t c u r r e n c i e s a r e q u o t e d a g a i n s t t h e U . S . d o l l a r , s o t o c a l c u l a t e t h e e x c h a n g e r a t e b e t w e e n c u r r e n c i e s o t h e r t h a n t h e U S $ n e e d t o c a l c u l a t e a c u r r e n c y c r o s s - r a t e .

T o d o t h i s c o r r e c t l y :

1 . E s t a b l i s h w h i c h c u r r e n c y i s t h e h o m e c u r r e n c y a n d w h i c h i s t h e f o r e i g n c u r r e n c y . 2 . P u t e x c h a n g e r a t e s i n t e r m s o f t h i r d c u r r e n c y i n d e n o m i n a t o r . 3 . D i v i d e o n e e x c h a n g e r a t e b y t h e o t h e r t o e l i m i n a t e t h e t h i r d c u r r e n c y .

J a p a n e s e Y e n : ¥ 1 0 5 . 2 5 / U S $ K o r e a n W o n : W 1 1 4 6 . 2 5 / U S $

W H A T I S T H E C R O S S R A T E B E T W E E N T H E Y E N A N D T H E W O N ?

C r o s s R a t e = J a p a n e s e Y e n / U S $

K o r e a n W o n / U S $

Y e n 1 0 5 . 2 5 / U S $

W 1 1 4 6 . 2 5 / U S $ ¥ 0 . 0 9 2 / W o n

M o r e c o m p l i c a t e d e x a m p l e s o f c r o s s - r a t e s t h a t t a k e i n t o a c c o u n t t h e a p p r o p r i a t e b i d o r a s k r a t e s i n t h e t r i a n g l e a r e g i v e n i n F I G S . 1 & 2 - B I D A N D A S K C R O S S R A T E S .

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Fig. 1 - Easy Cross Rate (no Bid/Ask)

JapanJapanYen

KoreaKoreaWon

$US

1. Buy US$ in Korea atWon Rate for $ =W 1146.25/US$

Divided byW 1146.25/US$

2. Sell $ in Tokyo at Yen rate for $ =

¥105.25/US$

Multiply by¥105.25/US$

Cross RateCross Rate¥0.092/Won

Result is Cross rate for Won in Japan

1 US$ buys about 1150 Won or 105 Yen … so 1 Won should buy about 1/10 of a Yen.

KNOW WHAT TO EXPECT AS A CHECK!!!!!KNOW WHAT TO EXPECT AS A CHECK!!!!!

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Cross Bid Rate – Rate at which Bank in Home buysbuys foreign currency– Your Alternative?

Sell Foreign currency for US$ Foreign Ask (Foreign/US$) Sell US$ for Home currency Home Bid (Home/US$) Your Exchange Rate? CCross BBid = Home BBid/Foreign AAsk

Cross Ask Rate – Rate at which Bank in Home sellssells foreign currency– Your Alternative?

Sell Home currency for US$ Home Ask (Home/US$) Sell US$ for Foreign currency Foreign Bid (Foreign/US$) Your Exchange Rate? CCross AAsk = Home AAsk/Foreign BBid

1. Cross rates NEVER have Bid/Bid or Ask /Ask !!!!!2. You always get the worst rate of the two possible!!!!

Cross Rates with Bid/Ask Spreads

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Fig. 2 - Cross Rates with Bid/Ask Spreads

BangkokBangkokBaht

BrazilBrazilReals

$US

One leg involves exchanging either

$ for Reals or Reals for $ $ is the foreign currency!

Bid -- 0.9955 Real/US$Ask -- 1.0076 Real/US$

Bid -- B 25.2513/US$Ask – B 25.3986/US$

Cross Bid RateCross Bid RateCross Ask RateCross Ask Rate

Results are Cross rates for Real in Bangkok

One leg involves exchanging either

$ for Baht or Baht for $ $ is the foreign currency!

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Fig. 3 - Bid Cross Rate

BangkokBangkokBaht

BrazilBrazilReals

$US

1. You buy US$ in Brazil Bank sells you US$ atReal ASK Rate for $ =

1.0076 Real/US$

Divided by1.0076 Real/US$

2. You sell $ in BangkokBank buys US$ from you at Baht BID rate for $ =

Baht 25.2513/US$

Multiply byBaht 25.2513/US$

Cross Bid RateCross Bid RateB25.0608/Real

Result is Bid cross rate for Real in Bangkok

Bid -- B 25.2513/US$Ask – B 25.3986/US$

Bid -- 0.9955 Real/US$Ask -- 1.0076 Real/US$

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Fig. 4 - Ask Cross Rate

BangkokBangkokBaht

BrazilBrazilReals

$US

2. You sell $ in Brazil Bank buys US$ at

Real BID Rate for $ =0.9955 Real/US$

Divided by0.9955 Real/US$

1. You buy $ in BangkokBank sells you US$ atBaht ASK rate for $ =

Baht 25.3986/US$

Multiply byBaht 25.3986/US$

Cross Ask RateCross Ask RateB25.5134/Real

Result is Ask cross rate for Real in Bangkok

Bid -- B 25.2513/US$Ask – B 25.3986/US$

Bid -- 0.9955 Real/US$Ask -- 1.0076 Real/US$

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Triangular Currency Arbitrage You should be able to calculate the profit or loss on triangular

arbitrage opportunity given three currency quotations; See next slide for numerical example. Set out triangle relating the three currencies & fill in the

exchange rates on appropriate side of the triangle. Set home currency (HC) & target foreign currency (TFC).

• Strategy 1: Move 1 unit of HC directly to TFC using exchange rate.• Strategy 2: Move 1 unit of HC indirectly through the second foreign

currency to get the TFC. Compare the amount of TFC from Strategy 1 vs. Strategy 2.

If they are not equal an arbitrage opportunity exists. • If TFC from Strategy 1 > TFC from Strategy 2. Then sell 1 unit HC

through Strategy 1, and use resulting TFC to buy HC with Strategy 2.• Result: More HC than began with, so make arbitrage profit.

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Triangular Currency ArbitrageStartStart

$1,000,000

2. Sell these DM in London at £1 = DM 3.1650 receive £ 505,448.35

3. Sell the pounds inNew York at £1 = $1.9809 receive

$1,001,242.64

New YorkNew York

1. Sell $1,000,000 inFrankfurt at DM 1

= $0.6251 receiveDM 1,599,744.04

DM 1,599,744.04

Divided by$0.6251/DM

FinishFinish$1,001,242.64

Multiply by$1.9809/ £

£ 505,448.35 Divided by

DM 3.1650/£LondonLondon FrankfurtFrankfurt

Arbitrage Profit!

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Forward FX Market

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Forward FX Market Forward foreign exchange contract calls for

delivery, at a fixed future date, of a specified amount of one currency against US$ payment. Exchange rate fixed by the forward contract is called the forward rate or the outright rate.

Difference between forward rate and current spot rate is the swap rate. There is a forward premium if the forward rate quoted in dollars is above the spot rate. There is a forward discount if forward rate is below the current spot rate.

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Forward Premium/Discount You should be able to calculate a forward discount or

premium and express it as an annualized rate; Annualized forward premium or discount to current spot

rate adjusts % difference between forward and spot rate for length of forward contract.

Interest rate parity theory ensures that return on a hedged foreign exchange rate position is just equal to the domestic interest rate on an investment of identical risk. If the returns are not identical then an arbitrage opportunity exists, and capital will flow to take advantage of the mispricing.

Forward Premium

or DiscountForward Rate - Spot Rate

Spot Rate

360

# days in Forward Contract

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Covered Interest Parity Condition

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Domestic vs Foreign Investment You are an U.S. investor seeking a one-year, risk-free

return on your money. You have two options:– Option 1: Invest in a one-year U.S. Gov’t Treasury Bill.

You earn the nominal interest rate.

– Option 2: Invest in a one-year Japanese Gov’t Treasury Bill. This involves three steps

Convert U.S. $ to Japanese Yen today. Invest Yen in Japan and earn the nominal interest rate. Convert the resulting Yen back into U.S.$ in one year’s time.

Option 1 is risk-free but Option 2 is not unless enter into forward today to sell the yen in one year.– Option 2 risk-free with forward – Covered position– Option 2 has risk without forward – Uncovered position.

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Covered Interest Parity

New YorkNew York

TokyoTokyo

$1$1

t = 0

t = 0 t = 1

t = 1

STRATEGY 1: Invest in US gov’t bond, earn (1 + iH)$(1 + i$(1 + iHH))

Invest in Japanese gov’t bond, earn (1 +iF)

¥¥(1+i(1+iFF)/e)/e00¥¥1/e1/e00

Strategy 2:Convert to Yen

at e0

Enter forward contract today to sell Yen at f1

$(1+i$(1+iFF)f)f11/e/e00

Both strategies are a riskless way of investing for one-period. Both strategies must have same return or arbitrage profit available. This is Covered Interest Parity condition

(1 + iH) = (1+ iF)f1/e0 OR f f11 = (1+ i = (1+ iHH)e)e00/(1 + i/(1 + iFF))

Strategies to invest $1 for one-period, want $ returns.

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S t r a t e g y 1 : D o m e s t i c I n v e s t m e n t – I n v e s t i n h o m e g o v ’ t b o n d y i e l d i n g n o m i n a l i n t e r e s t r a t e , i H .

R e t u r n i n H C i s = ( 1 + i H )

S t r a t e g y 2 : H e d g e d F o r e i g n I n v e s t m e n t - C o n v e r t H C a t c u r r e n t e x c h a n g e r a t e , e 0 , i n v e s t i n F o r e i g n g o v ’ t b o n d y i e l d i n g r F , c o n v e r t p r o c e e d s b a c k t o H C a t f o r w a r d r a t e , f 1 .

R e t u r n i n H C i s = 1

0

1 Fi f

e

A r b i t r a g e P r o f i t s a v a i l a b l e i f 1

0

11 F

H

i fi

e

H o w t o t a k e a d v a n t a g e o f t h i s m i s p r i c i n g ? L o o k a t t h e r e t u r n s i n p a r t k . B o r r o w H o m e C u r r e n c y l o w a n d l e n d i t l e n d h i g h .

I f 1

0

11 F

H

i fi

e

, b o r r o w i n h o m e c o u n t r y , l e n d ( h e d g e d f o r e i g n i n v e s t m e n t )

f o r e i g n .

I f 1

0

11 F

H

i fi

e

, b o r r o w i n f o r e i g n c o u n t r y ( h e d g e d f o r e i g n b o r r o w i n g ) , l e n d

h o m e .

B e c a r e f u l ! T h e i n t e r e s t r a t e s m u s t m a t c h t h e f o r w a r d c o n t r a c t i n d u r a t i o n !

Covered Interest Parity Condition Covered foreign investment returns should equal

domestic investment returns under arbitrage.

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Covered Interest Parity ArbitrageNew YorkNew York

LondonLondon

t = 0

t = 0 t = 1

t = 1

$1,000,000$1,000,000

1: Borrow $1,000,000 at 7% for 1 year. OweOwe$1,070,000$1,070,000

3. Invest in London, earn 12% for 1 year£640,000£640,000££571,428.57571,428.57

2. Convert $ to £at e0 = $1.75/ £ 4. Sell £ forward

at f1= $1.68/£

ReceiveReceive$1,075,200$1,075,200

Arbitrage Profit = $,1075,200 - $1,070,000 = $5,200Arbitrage Profit = $,1075,200 - $1,070,000 = $5,200 Note that transaction costs (bid-ask spreads, etc.) will reduce these profits.

Arbitrage Profit!Arbitrage Profit!

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Factors Hindering CIPC - in the real world, a variety of things may interfere

with ability of arbitrageurs to attain CIPC.

– Transaction CostsTransaction Costs: empirically very small for most traded currencies.

– Costs of InformationCosts of Information: again, empirically very small.

– Government Intervention and RegulationGovernment Intervention and Regulation: varies by currency and time period.

– Financial Constraints/Capital market ImperfectionsFinancial Constraints/Capital market Imperfections

– Noncomparable AssetsNoncomparable Assets differing default rates & political risk. require premia in CIPC to offset risk.

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Covered Interest Arbitrage Parity Line

iNY - iLondon (+)

iNY - iLondon (-)

Forward Discount, p (-)

Forward Premium, p (+)

CIAP Line

Transaction Costs

Invest Homei > i* + p

Invest Foreigni < i* + p

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Market Adjustments to CIAPiLondon

£’s

S£

D£

S$

D$

S£

D£

S£

D£

iNY

$’sLondon Money Market

Forward FX MarketSpot FX Market

NY Money Market

£’s £’s

e $/£f1 $/£

S’£S’$

S’£

D’£

(1 + iH) > (1+ iF)f1/e0(1 + iH) = (1+ iF)f1/e0

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Uncovered Interest Parity Condition

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Domestic vs Foreign Investment You are a risk-neutral U.S. investor seeking highest

one-year expected return on your money. – Option 1: Invest in a one-year U.S. Gov’t Treasury Bill.

You earn the nominal interest rate.

– Option 2: Invest in a one-year Japanese Gov’t Treasury Bill. This involves three steps

Convert U.S. $ to Japanese Yen today. Invest Yen in Japan and earn the nominal interest rate. Convert the resulting Yen back into U.S.$ in one year’s time.

Risk-neutrality implies that you care only about expected return, not risk. – Uncovered Foreign investment should have same expected

return as Domestic investment or expected arbitrage profits.

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Uncovered Interest Parity

New YorkNew York

TokyoTokyo

$1$1

t = 0

t = 0 t = 1

t = 1

STRATEGY 1: Invest in US gov’t bond, earn (1 + rH)$(1 + i$(1 + iHH))

Invest in Japanese gov’t bond, earn (1 + iF)

¥¥(1+i(1+iFF)/e)/e00¥¥1/e1/e00

Strategy 2:Convert to Yen

at e0

Convert Yen at expected EXR in one year, eE

1

$(1+i$(1+iFF)e)eEE11/e/e00

Uncovered foreign investment involves future EXR risk for one-period. For risk-neutral investor both strategies should have same expected return. This is Uncovered Interest Parity condition

(1 + iH) = (1+ iF)eE1/e0 OR e eEE

11// ee00 = (1+ i= (1+ iHH) /(1 + i) /(1 + iFF))

Strategies to invest $1 for one-period, want $ returns.

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S t r a t e g y 1 : D o m e s t i c I n v e s t m e n t – I n v e s t i n h o m e g o v ’ t b o n d y i e l d i n g n o m i n a l i n t e r e s t r a t e , i H .

R e t u r n i n H C i s = ( 1 + i H )

S t r a t e g y 2 : U n c o v e r e d F o r e i g n I n v e s t m e n t - C o n v e r t H C a t c u r r e n t e x c h a n g e r a t e , e 0 , i n v e s t i n F o r e i g n g o v ’ t b o n d y i e l d i n g r F , c o n v e r t p r o c e e d s b a c k t o H C a t f u t u r e e x c h a n g e r a t e r a t e , e E

1 .

R e t u r n i n H C i s = 1

0

1 EFi e

e

A r b i t r a g e P r o f i t s a v a i l a b l e i f 1

0

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EF

H

i ei

e

H o w t o t a k e a d v a n t a g e o f t h i s m i s p r i c i n g ? L o o k a t t h e r e t u r n s . B o r r o w H o m e C u r r e n c y l o w a n d l e n d i t l e n d h i g h .

I f 1

0

11

EF

H

i ei

e

, b o r r o w i n h o m e c o u n t r y , l e n d ( u n c o v e r e d f o r e i g n i n v e s t m e n t ) f o r e i g n .

I f 1

0

11

EF

H

i ei

e

, b o r r o w i n f o r e i g n c o u n t r y ( u n c o v e r e d f o r e i g n b o r r o w i n g ) , l e n d h o m e .

B e c a r e f u l ! T h e i n t e r e s t r a t e s m u s t m a t c h t h e i n v e s t m e n t i n d u r a t i o n !

Uncovered Interest Parity Condition

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Uncovered Interest Arbitrage Parity Line

iNY - iLondon (+)

iNY - iLondon (-)

Expected Foreign Depreciation

Expected Foreign Appreciation

UIAP Line

Transaction Costs