Introduction to Exchange Rate Calculations
Mastering exchange rate calculations is crucial for the CFA exam and practical application in international finance. This chapter provides step-by-step approaches to solve various types of exchange rate problems.
Key Calculation Types
Spot Rate Conversions
Converting between currencies using current exchange rates, including bid-ask spread considerations.
Cross Rate Calculations
Determining exchange rates between two currencies when only their rates against a third currency are known.
Forward Rate Calculations
Computing future exchange rates using interest rate parity and spot rates.
Investment Return Calculations
Calculating total returns on foreign investments including currency gains/losses.
Basic Exchange Rate Calculations
Understanding fundamental conversion methods forms the foundation for more complex calculations.
Currency Conversion Formula
Example 1: Basic Currency Conversion
Problem: Convert $10,000 USD to EUR given EUR/USD = 1.0850
Step 1: Identify the conversion needed
We need to convert from USD to EUR. The rate given is EUR/USD = 1.0850, meaning 1 EUR = 1.0850 USD.
Step 2: Set up the calculation
EUR Amount = USD Amount ÷ EUR/USD rate
EUR Amount = $10,000 ÷ 1.0850
Step 3: Calculate the result
EUR Amount = €9,216.59
Bid-Ask Spread Considerations
In real markets, there are different rates for buying and selling currencies.
| Transaction Type | Rate Used | Explanation |
|---|---|---|
| Buying Base Currency | Ask Rate | Higher rate - bank's selling price |
| Selling Base Currency | Bid Rate | Lower rate - bank's buying price |
Example 2: Bid-Ask Spread Calculation
Given: EUR/USD Bid = 1.0849, EUR/USD Ask = 1.0851
Problem: Calculate the cost to convert $100,000 to EUR and immediately back to USD
Step 1: Convert USD to EUR (use ask rate)
EUR = $100,000 ÷ 1.0851 = €92,161.46
Step 2: Convert EUR back to USD (use bid rate)
USD = €92,161.46 × 1.0849 = $99,981.62
Step 3: Calculate the transaction cost
Cost = $100,000 - $99,981.62 = $18.38
Cross Rate Calculations
Cross rates allow us to find exchange rates between two currencies using their rates against a common third currency (typically USD).
Cross Rate Formulas
Direct Cross Rate
When both currencies are quoted in terms of USD
Inverse Cross Rate
When one currency is quoted against USD and the other has USD in numerator
Example 3: Calculating EUR/JPY Cross Rate
Given:
- EUR/USD = 1.0850
- USD/JPY = 110.25
Step 1: Identify the calculation method
We have EUR/USD and USD/JPY. To find EUR/JPY, we multiply these rates.
Step 2: Apply the cross rate formula
EUR/JPY = EUR/USD × USD/JPY
EUR/JPY = 1.0850 × 110.25
Step 3: Calculate the result
EUR/JPY = 119.62
Interpretation: 1 EUR = 119.62 JPY
Example 4: Calculating GBP/CHF Cross Rate
Given:
- GBP/USD = 1.3625
- USD/CHF = 0.9180
Step 1: Set up the calculation
GBP/CHF = GBP/USD × USD/CHF
Step 2: Calculate
GBP/CHF = 1.3625 × 0.9180 = 1.2508
Step 3: Verify the logic
1 GBP = 1.3625 USD, and 1 USD = 0.9180 CHF
Therefore: 1 GBP = 1.2508 CHF ✓
Forward Rate Calculations
Forward rates are calculated using interest rate parity theory and current spot rates.
Interest Rate Parity Formula
Where:
rd = domestic interest rate
rf = foreign interest rate
t = time to maturity in days
Example 5: 6-Month EUR/USD Forward Rate
Given:
- Current EUR/USD spot rate: 1.0850
- US 6-month interest rate: 2.75%
- Eurozone 6-month interest rate: 1.25%
- Time period: 180 days
Step 1: Identify domestic and foreign rates
For EUR/USD from US perspective: USD is domestic (rd = 2.75%), EUR is foreign (rf = 1.25%)
Step 2: Apply the forward rate formula
Forward Rate = 1.0850 × [(1 + 0.0275 × 180/360) / (1 + 0.0125 × 180/360)]
Forward Rate = 1.0850 × [(1 + 0.01375) / (1 + 0.00625)]
Forward Rate = 1.0850 × [1.01375 / 1.00625]
Step 3: Calculate the result
Forward Rate = 1.0850 × 1.00745 = 1.0931
Result: 6-month EUR/USD forward rate = 1.0931
Forward Premium/Discount Calculation
Example 6: Forward Premium Calculation
Using the result from Example 5:
Spot Rate = 1.0850, Forward Rate = 1.0931
Calculate the forward premium
Premium = [(1.0931 - 1.0850) / 1.0850] × 100%
Premium = [0.0081 / 1.0850] × 100% = 0.747%
Result: EUR trades at a 0.747% forward premium (annualized: 1.494%)
Foreign Investment Return Calculations
When investing in foreign assets, total return includes both the asset return and currency effects.
Total Return Formula
OR
Total Return = (1 + Asset Return) × (1 + Currency Return) - 1
Example 7: US Investment in European Stocks
Scenario: US investor buys European stocks at the beginning of the year
Given:
- Initial EUR/USD rate: 1.0800
- Final EUR/USD rate: 1.1200
- Stock return in EUR: +12%
Step 1: Calculate currency return
Currency Return = (Final Rate - Initial Rate) / Initial Rate
Currency Return = (1.1200 - 1.0800) / 1.0800 = 0.04 / 1.0800 = 3.70%
Step 2: Calculate total return
Total Return = 0.12 + 0.037 + (0.12 × 0.037)
Total Return = 0.12 + 0.037 + 0.00444 = 0.1614
Step 3: Final result
Total Return in USD: +16.14%
Asset return: +12.00%, Currency return: +3.70%, Interaction: +0.44%
Example 8: Currency Hedged vs. Unhedged Returns
Scenario: Compare hedged and unhedged investment returns
Given:
- Japanese stock return: +8% (in JPY)
- USD/JPY initial: 110.00
- USD/JPY final: 108.50
- Forward rate used for hedging: 109.75
Unhedged Return Calculation
Currency Return = (108.50 - 110.00) / 110.00 = -1.36%
Total Return = 0.08 + (-0.0136) + (0.08 × -0.0136) = 6.53%
Hedged Return Calculation
With currency hedging, only asset return applies: +8.00%
The forward contract eliminates currency risk
Comparison
Unhedged return: +6.53% | Hedged return: +8.00%
Hedging added +1.47% by eliminating negative currency impact
Practice Problems
Test your understanding with these CFA-style exchange rate calculation problems.
Practice Problem 1: Cross Rate Arbitrage
Given market quotes:
- EUR/USD = 1.0850
- GBP/USD = 1.3625
- EUR/GBP (quoted) = 0.7950
Question: Is there an arbitrage opportunity? Calculate the profit from trading €1,000,000.
Step 1: Calculate implied EUR/GBP cross rate
Implied EUR/GBP = EUR/USD ÷ GBP/USD = 1.0850 ÷ 1.3625 = 0.7963
Step 2: Compare with market quote
Market quote: 0.7950 | Implied rate: 0.7963
EUR is undervalued in the cross rate → Buy EUR/GBP, Sell EUR/USD, Buy GBP/USD
Step 3: Execute arbitrage
1. Convert €1,000,000 → $1,085,000 (sell EUR/USD)
2. Convert $1,085,000 → £796,330 (buy GBP/USD at 1.3625)
3. Convert £796,330 → €1,001,634 (sell EUR/GBP at 0.7950)
Profit: €1,634
Practice Problem 2: Multi-Period Forward Rate
Given:
- Current AUD/USD spot: 0.7250
- Australian 1-year rate: 3.50%
- US 1-year rate: 2.25%
Question: Calculate the 1-year forward AUD/USD rate and determine if AUD trades at a premium or discount.
Solution:
Forward = 0.7250 × [(1 + 0.0225) / (1 + 0.035)]
Forward = 0.7250 × [1.0225 / 1.035] = 0.7250 × 0.9879 = 0.7162
Premium/Discount = (0.7162 - 0.7250) / 0.7250 = -1.21%
Result: AUD trades at a 1.21% forward discount
- Always check currency conventions and quote directions
- Use dimensional analysis to verify your calculations
- Remember that higher interest rate currencies typically trade at forward discounts
- Practice converting between different quote conventions (direct/indirect)