Introduction to Exchange Rates and the Foreign Exchange Market

Essentials of Exchange Rates

  • Definition of an Exchange Rate (EE): An exchange rate is the price of a foreign currency expressed in terms of a home (domestic) currency. It represents the relative price of two currencies.

  • Quotation Methods: To avoid confusion, the home and foreign countries must be specified. Two ways of quoting exist:

    • Home Currency Units per Foreign Unit: The number of home currency units exchanged for one unit of foreign currency.

    • Foreign Currency Units per Home Unit: The number of foreign currency units exchanged for one unit of home currency.

  • Standard Notation (E1/2E_{1/2}): This denotes the exchange rate of country 1, expressed in units of country 1’s currency per one unit of country 2’s currency.

    • United States Example: The U.S. exchange rate with the Eurozone is denoted as E$/‐E_{\$/‐} (U.S. dollars per euro).

    • Denmark Example: Denmark’s exchange rate with the Eurozone is denoted as Ekr/‐E_{kr/‐} (Danish krone per euro).

  • Reciprocal Relationship: The exchange rate in one set of terms is the mathematical inverse of the other. For example:

    • E$/‐=1E‐/$E_{\$/‐} = \frac{1}{E_{‐/\$}}

    • Using November 15, 2019 data: 1.1045=10.90541.1045 = \frac{1}{0.9054}.

Exchange Rate Quotations (November 15, 2019 vs. November 15, 2018)

  • Data Table Analysis: The following major exchange rates compare 2019 values Against 2018 (one year prior):

    • Canada (Dollar, C\): 2019 Per 1 $=1.32301\,\$ = 1.3230; 2019 Per 1 ‐=1.46131\,‐ = 1.4613; 2018 Per 1 $=1.31921\,\$ = 1.3192.

    • Denmark (Krone, DKrDKr): 2019 Per 1 $=6.76611\,\$ = 6.7661; 2019 Per 1 ‐=7.47311\,‐ = 7.4731; 2018 Per 1 $=6.58901\,\$ = 6.5890.

    • Eurozone (Euro, ‐‐): 2019 Per 1 $=0.90541\,\$ = 0.9054 (European terms); 2018 Per 1 $=0.88301\,\$ = 0.8830.

    • Japan (Yen, 3): 2019 Per 1 $=108.811\,\$ = 108.81; 2019 Per 1 ‐=120.171\,‐ = 120.17; 2018 Per 1 $=113.451\,\$ = 113.45.

    • Norway (Krone, NKrNKr): 2019 Per 1 $=9.08701\,\$ = 9.0870; 2019 Per 1 ‐=10.03661\,‐ = 10.0366; 2018 Per 1 $=8.47901\,\$ = 8.4790.

    • Sweden (Krona, SKrSKr): 2019 Per 1 $=9.64891\,\$ = 9.6489; 2019 Per 1 ‐=10.65721\,‐ = 10.6572; 2018 Per 1 $=9.05191\,\$ = 9.0519.

    • Switzerland (Franc, SFrSFr): 2019 Per 1 $=0.99041\,\$ = 0.9904; 2019 Per 1 ‐=1.09391\,‐ = 1.0939; 2018 Per 1 $=1.00561\,\$ = 1.0056.

    • United Kingdom (Pound, 3): 2019 Per 1 $=0.77521\,\$ = 0.7752; 2019 Per 1 ‐=0.85621\,‐ = 0.8562; 2018 Per 1 $=0.78411\,\$ = 0.7841.

    • United States (Dollar, \): 2019 Per 1 ‐=1.10451\,‐ = 1.1045 (American terms); 2019 Per 1\,3 = 1.2900; 2018 Per 1 ‐=1.13251\,‐ = 1.1325.

Appreciations and Depreciations

  • Definitions:

    • Appreciation: If one currency buys more of another currency, it has experienced an appreciation (its value has risen).

    • Depreciation: If a currency buys less of another currency, it has experienced a depreciation (its value has fallen).

  • Dynamics in U.S. Terms (E$/‐E_{\$/‐}):

    • When the exchange rate E$/‐E_{\$/‐} rises, the price of one euro increases in dollar terms; therefore, the U.S. dollar depreciates.

    • When the exchange rate E$/‐E_{\$/‐} falls, the price of one euro decreases in dollar terms; therefore, the U.S. dollar appreciates.

  • Dynamics in European Terms (E‐/$E_{‐/\$}):

    • When the exchange rate E‐/$E_{‐/\$} rises, the price of one dollar increases in euro terms; the euro depreciates.

    • When the exchange rate E‐/$E_{‐/\$} falls, the price of one dollar decreases in euro terms; the euro appreciates.

  • Calculating Proportional Change:

    • To determine size, compute the percentage change: ΔEtEt\frac{\Delta E_{t}}{E_{t}}.

    • Example (US Dollar appreciation):

      • E$/‐,t(2018)=$1.1325E_{\$/‐, t} (2018) = \$1.1325

      • E$/‐,t+1(2019)=$1.1045E_{\$/‐, t+1} (2019) = \$1.1045

      • ΔE$/‐,t=1.1045−1.1325=−$0.0280\Delta E_{\$/‐, t} = 1.1045 - 1.1325 = -\$0.0280

      • Percentage Change=−0.02801.1325=−2.47%\text{Percentage Change} = \frac{-0.0280}{1.1325} = -2.47\%

      • Result: The dollar appreciated against the euro by 2.47%2.47\%.

    • Example (Euro depreciation):

      • E‐/$,t(2018)=‐0.8830E_{‐/\$, t} (2018) = ‐0.8830

      • E‐/$,t+1(2019)=‐0.9054E_{‐/\$, t+1} (2019) = ‐0.9054

      • ΔE‐/$,t=0.9054−0.8830=+‐0.0224\Delta E_{‐/\$, t} = 0.9054 - 0.8830 = +‐0.0224

      • Percentage Change=0.02240.8830=+2.53%\text{Percentage Change} = \frac{0.0224}{0.8830} = +2.53\%

      • Result: The euro depreciated against the dollar by 2.53%2.53\%.

Multilateral Exchange Rates

  • Effective Exchange Rate: This is a measure of the average change in a currency's value against a basket of foreign currencies, aggregated using trade weights.

  • Mathematical Formula: For NN currencies in a basket, where Total Trade = Trade1+Trade2+⋯+TradeN\text{Trade}_{1} + \text{Trade}_{2} + \dots + \text{Trade}_{N}, the change in the home country's effective exchange rate (EeffectiveE_{\text{effective}}) is:

    • ΔEeffectiveEeffective=∑i=1N(TradeiTrade×ΔEiEi)\frac{\Delta E_{\text{effective}}}{E_{\text{effective}}} = \sum_{i=1}^{N} \left( \frac{\text{Trade}_{i}}{\text{Trade}} \times \frac{\Delta E_{i}}{E_{i}} \right)

  • Calculation Example:

    • Assume: 40%40\% of Home trade is with Country 1; 60%60\% is with Country 2.

    • Home currency appreciates 10%10\% against Country 1 but depreciates 30%30\% against Country 2.

    • Change=(−10%×0.40)+(+30%×0.60)\text{Change} = (-10\% \times 0.40) + (+30\% \times 0.60)

    • Change=−0.04+0.18=+0.14=+14%\text{Change} = -0.04 + 0.18 = +0.14 = +14\%

    • Result: Home’s effective exchange rate depreciated by 14%14\%.

  • U.S. Dollar Trends (2002–2019):

    • Against a basket of 7 major currencies, the dollar depreciated by 35%35\% by early 2008.

    • Against a broad basket of 26 currencies, it lost only 25%25\% by 2008.

    • Reason for Discrepancy: The dollar floated against major currencies, but the broad basket included partners (like China) that maintained fixed or tightly managed rates against the dollar.

The Impact of Exchange Rates on Prices

  • Hypothetical Case Study: James Bond's Tuxedo:

    • Prices in Local Currency: London (0˘01232,000\u00123 2,000), Hong Kong (HK$30,000HK\$ 30,000), New York ($4,000\$4,000).

    • In Scenario 4, where EHK$/0˘0123=14E_{HK\$/\u00123} = 14 and E$/0˘0123=1.9E_{\$/\u00123} = 1.9:

      • Tuxedo cost in London = 0˘01232,000\u00123 2,000

      • Tuxedo cost in Hong Kong (in pounds) = 30,000/14≈0˘01232,14330,000 / 14 \approx \u00123 2,143

      • Tuxedo cost in New York (in pounds) = 4,000/1.9≈0˘01232,1054,000 / 1.9 \approx \u00123 2,105

  • Generalizations (Holding Local Prices Constant):

    • Changes in exchange rates cause proportional changes in the prices of foreign goods expressed in home currency.

    • Depreciation Effects: Home exports become less expensive to foreigners; foreign exports become more expensive to home residents.

    • Appreciation Effects: Home exports become more expensive to foreigners; foreign exports become less expensive to home residents.

Exchange Rate Regimes

  • Fixed (Pegged) Exchange Rate: Fluctuates in a narrow range (or not at all) against a base currency. Maintenance requires government intervention in the foreign exchange market.

    • Band: A fixed regime with tiny variations (e.g., Danish krone against the euro fluctuates within a band of ±2%\pm 2\%).

    • Currency Board: An ultra-hard peg with special legal and procedural rules to make the peg more durable (6 countries used this in 2010).

  • Floating (Flexible) Exchange Rate: Fluctuates in a wider range; the government does not attempt to fix it. These are subject to high volatility (e.g., U.S. dollar vs. yen, pound, or loonie).

  • Managed Float: Middle-ground regimes like crawling pegs.

    • Crawling Peg: The currency is allowed to depreciate steadily at an almost constant rate (e.g., Colombian peso, 1996–2002).

  • Dollarization: Occurs when a country unilaterally adopts the currency of another country (e.g., Ecuador in 2000).

  • IMF Classification (2010): Out of 182 economies:

    • 66: Currency boards

    • 3535: Hard pegs

    • 4343: Bands, crawling pegs, or crawling bands

    • 4646: Free floats, managed floats, or wide bands

The Foreign Exchange (Forex/FX) Market

  • Market Structure: Not an organized exchange; it is conducted "over the counter" (OTC) in multiple worldwide locations.

  • Market Size: In April 2019, global trade reached $6.6 trillion\$6.6\,\text{trillion} per day. This is a 29% increase from 2016 and over six times the volume of 1992.

  • Major Trading Centers: London, New York, Singapore, and Hong Kong (Tier 1). Also significant: Tokyo, Zurich, Paris, and Frankfurt.

  • Contract Types:

    • Spot Contract: The simplest transaction; an immediate exchange of one currency for another. The rate is the "spot exchange rate."

    • Derivatives: Related contracts including Forwards, Swaps, Futures, and Options.

  • Transaction Costs (The Spread): The difference between the "buy at" (bid) and "sell for" (ask) prices.

    • Retail transactions: 2%2\%–5%5\%.

    • Large banks/firms: Less than 0.10%0.10\% (major currencies range from 0.01%0.01\% to 0.03%0.03\%).

  • Private and Government Actors:

    • Commercial Banks: Handle more than half of global transactions (Top 10 banks).

    • Capital Controls: Government policies that restrict the trading or movement of forex.

Arbitrage and No-Arbitrage Conditions

  • Bilateral Arbitrage: Ensures that the exchange rate between two points (e.g., NY and London) is the same regardless of the path taken.

  • Triangular Arbitrage (Three Currencies): To avoid profit opportunities, the direct exchange rate must equal the cross rate.

    • E0˘0123/$=E0˘0123/‐×E‐/$=E0˘0123/‐×1E$/‐E_{\u00123/\$} = E_{\u00123/‐} \times E_{‐/\$} = E_{\u00123/‐} \times \frac{1}{E_{\$/‐}}

  • Vehicle Currency: A third currency (often the U.S. dollar) used for intermediation between two other currencies because of its high liquidity and widespread use.

Arbitrage and Interest Rates (Parity Conditions)

  • Asset Attributes: Investors consider Return (net wealth increase), Risk (volatility of return), and Liquidity (ease of liquidation).

  • Covered Interest Parity (CIP): Riskless arbitrage using a forward contract to eliminate exchange rate risk.

    • Dollar Return on Dollar Deposit: (1+i$)(1 + i_{\$})

    • Dollar Return on Euro Deposit (Covered): F$/‐E$/‐(1+i‐)\frac{F_{\$/‐}}{E_{\$/‐}} (1 + i_{‐})

    • CIP Equation: (1+i$)=F$/‐E$/‐(1+i‐)(1 + i_{\$}) = \frac{F_{\$/‐}}{E_{\$/‐}} (1 + i_{‐})

  • Uncovered Interest Parity (UIP): Risky arbitrage where traders face exchange rate risk and use the expected future spot rate (E$/‐eE^{e}_{\$/‐}).

    • UIP Equation: (1+i$)=E$/‐eE$/‐(1+i‐)(1 + i_{\$}) = \frac{E^{e}_{\$/‐}}{E_{\$/‐}} (1 + i_{‐})

    • Spot Rate Determination: Based on UIP, the spot rate can be solved as E$/‐=E$/‐e×1+i‐1+i$E_{\$/‐} = E^{e}_{\$/‐} \times \frac{1 + i_{‐}}{1 + i_{\$}}.

  • Evidence and Liberalization:

    • In the 1970s, large differences in returns existed between UK pound and German mark deposits due to capital controls.

    • After financial liberalization, these profits vanished, and CIP established itself as a standard condition.

    • If both CIP and UIP hold, then F$/‐=E$/‐eF_{\$/‐} = E^{e}_{\$/‐}.

UIP Approximation

  • Formula: i$≈i‐+ΔE$/‐eE$/‐i_{\$} \approx i_{‐} + \frac{\Delta E^{e}_{\$/‐}}{E_{\$/‐}}

  • Interpretation: The home interest rate equals the foreign interest rate plus the expected rate of depreciation of the home currency.

  • Example: If dollar interest is 4%4\% and euro interest is 3%3\%, UIP holds if the expected dollar depreciation is 1%1\%. The total dollar return on the euro deposit effectively matches the 4%4\% offered by dollar deposits.

Summary of Parity Relationships

  • Spot Market: UIP serves as a model for how the spot exchange rate is determined, requiring knowledge of prevailing interest rates and expected future exchange rates.

  • Forward Market: CIP serves as a model for how the forward exchange rate is determined, derived from the current spot rate and the interest rates of the two currencies.