International Economics

International Economics Foreign Exchange Calculations and Arbitrage

Spot FX Market

Structure of FX Market

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, PA: 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, PB: rate at which the bank will buy foreign currency in return for domestic currency, i.e. USD.38/DM (or 2.63 DM/USD) • SPREAD: 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 = [(PA - PB )/PA] x 100

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 = [PA + PB]/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

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.

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.

Currency Cross-Rates • Most currencies are quoted against the US$, so to get exchange rate between two currencies need to be able to calculate currency cross rates given two spot or forward foreign exchange quotations involving three currencies; • Set which currency is home currency & which is foreign currency. • Put exchange rates in terms of the third currency in denominator. • Divide one exchange rate by the other to eliminate the third currency. • Example Direct Cross Rate for Won in Japan: • Japanese Yen-US$ 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 = ¥105.25/US$ W1146.25/US$ Direct Cross Rate for Won in Japan = ¥0.092/Won • More complicated examples of cross-rates take into account the appropriate bid or ask rates.

Currency Cross Rates

Result is Cross rate for Won in Japan Cross Rate ¥0.092/Won 2. Sell $ in Tokyo at Yen rate for $ = ¥105.25/US$ 1. Buy US$ in Korea at Won Rate for $ = W 1146.25/US$ Divided by W 1146.25/US$ Multiply by ¥105.25/US$ Fig. 1 - Easy Cross Rate (no Bid/Ask) Korea Won Japan Yen 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!!!!! $US

Cross Rates with Bid/Ask Spreads • Cross Bid Rate • Rate at which Bank in Home buys foreign currency • Your Alternative? • Sell Foreign currency for US$ Foreign Ask (Foreign/US$) • Sell US$ for Home currency Home Bid (Home/US$) • Your Exchange Rate? Cross Bid = Home Bid/Foreign Ask • Cross Ask Rate • Rate at which Bank in Home sells foreign currency • Your Alternative? • Sell Home currency for US$ Home Ask (Home/US$) • Sell US$ for Foreign currency Foreign Bid (Foreign/US$) • Your Exchange Rate? Cross Ask = Home Ask/Foreign Bid • Cross rates NEVER have Bid/Bid or Ask /Ask !!!!! • You always get the worst rate of the two possible!!!!

Fig. 2 - Cross Rates with Bid/Ask Spreads Results are Cross rates for Real in Bangkok Cross Bid Rate Cross Ask Rate Brazil Reals Bangkok Baht One leg involves exchanging either $ for Baht or Baht for $ $ is the foreign currency! One leg involves exchanging either $ for Reals or Reals for $ $ is the foreign currency! Bid -- B 25.2513/US$ Ask – B 25.3986/US$ Bid -- 0.9955 Real/US$ Ask -- 1.0076 Real/US$ $US

Result is Bid cross rate for Real in Bangkok Cross Bid Rate B25.0608/Real 2. You sell $ in Bangkok Bank buys US$ from you at Baht BID rate for $ = Baht 25.2513/US$ 1. You buy US$ in Brazil Bank sells you US$ at Real ASK Rate for $ = 1.0076 Real/US$ Divided by 1.0076 Real/US$ Multiply by Baht 25.2513/US$ Fig. 3 - Bid Cross Rate Brazil Reals Bangkok Baht Bid -- B 25.2513/US$ Ask – B 25.3986/US$ Bid -- 0.9955 Real/US$ Ask -- 1.0076 Real/US$ $US

Result is Ask cross rate for Real in Bangkok Cross Ask Rate B25.5134/Real 1. You buy $ in Bangkok Bank sells you US$ at Baht ASK rate for $ = Baht 25.3986/US$ 2. You sell $ in Brazil Bank buys US$ at Real BID Rate for $ = 0.9955 Real/US$ Divided by 0.9955 Real/US$ Multiply by Baht 25.3986/US$ Fig. 4 - Ask Cross Rate Brazil Reals Bangkok Baht Bid -- B 25.2513/US$ Ask – B 25.3986/US$ Bid -- 0.9955 Real/US$ Ask -- 1.0076 Real/US$ $US

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.

Arbitrage Profit! Finish $1,001,242.64 3. Sell the pounds in New York at £1 = $1.9809 receive $1,001,242.64 1. Sell $1,000,000 in Frankfurt at DM 1 = $0.6251 receive DM 1,599,744.04 Multiply by $1.9809/ £ Divided by $0.6251/DM £ 505,448.35 DM 1,599,744.04 Divided by DM 3.1650/£ 2. Sell these DM in London at £1 = DM 3.1650 receive £ 505,448.35 Triangular Currency Arbitrage New York Start $1,000,000 London Frankfurt

Forward FX Market

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.

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.

Covered Interest Parity Condition

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.

STRATEGY 1: Invest in US gov’t bond, earn (1 + iH) $(1 + iH) $(1+iF)f1/e0 Strategy 2: Convert to Yen at e0 Enter forward contract today to sell Yen at f1 Invest in Japanese gov’t bond, earn (1 +iF) ¥1/e0 ¥(1+iF)/e0 Covered Interest Parity • Strategies to invest $1 for one-period, want $ returns. $1 New York t = 1 t = 0 Tokyo t = 0 t = 1 • 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/e0OR f1 = (1+ iH)e0/(1 + iF)

Covered Interest Parity Condition • Covered foreign investment returns should equal domestic investment returns under arbitrage.

Arbitrage Profit! Owe $1,070,000 1: Borrow $1,000,000 at 7% for 1 year. $1,000,000 Receive $1,075,200 2. Convert $ to £ at e0 = $1.75/ £ 4. Sell £ forward at f1=$1.68/£ £571,428.57 £640,000 3. Invest in London, earn 12% for 1 year Covered Interest Parity Arbitrage New York t = 0 t = 1 t = 0 t = 1 London • Arbitrage Profit = $,1075,200 - $1,070,000 = $5,200 • Note that transaction costs (bid-ask spreads, etc.) will reduce these profits.

Factors Hindering CIPC • - in the real world, a variety of things may interfere with ability of arbitrageurs to attain CIPC. • Transaction Costs: empirically very small for most traded currencies. • Costs of Information: again, empirically very small. • Government Intervention and Regulation: varies by currency and time period. • Financial Constraints/Capital market Imperfections • Noncomparable Assets • differing default rates & political risk. • require premia in CIPC to offset risk.

CIAP Line Invest Home i > i* + p Invest Foreign i < i* + p Transaction Costs Covered Interest Arbitrage Parity Line iNY - iLondon(+) Forward Discount, p (-) Forward Premium, p (+) iNY - iLondon (-)

S’£ S’$ S’£ D’£ Market Adjustments to CIAP iLondon iNY S$ S£ D£ D$ £’s $’s London Money Market NY Money Market (1 + iH) > (1+ iF)f1/e0 (1 + iH) = (1+ iF)f1/e0 e $/£ S£ f1 $/£ S£ D£ D£ Spot FX Market £’s Forward FX Market £’s

Uncovered Interest Parity Condition

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.

STRATEGY 1: Invest in US gov’t bond, earn (1 + rH) $(1 + iH) $(1+iF)eE1/e0 Strategy 2: Convert to Yen at e0 Convert Yen at expected EXR in one year, eE1 Invest in Japanese gov’t bond, earn (1 + iF) ¥1/e0 ¥(1+iF)/e0 Uncovered Interest Parity • Strategies to invest $1 for one-period, want $ returns. $1 New York t = 1 t = 0 Tokyo t = 0 t = 1 • 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/e0OR eE1/e0= (1+ iH) /(1 + iF)

Uncovered Interest Parity Condition

UIAP Line Transaction Costs Uncovered Interest Arbitrage Parity Line iNY - iLondon(+) Expected Foreign Depreciation Expected Foreign Appreciation iNY - iLondon (-)

International Economics