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Exchange rate Calculations

Economics

Cross-Rate Calculations

Learning Outcome Statement:

calculate and interpret currency cross-rates

Summary:

This LOS covers the calculation and interpretation of currency cross-rates in the foreign exchange (FX) market. It explains how to derive exchange rates between two currencies using a third currency and discusses the conventions used in the FX market for quoting exchange rates. The content also addresses the process of inverting exchange rates when necessary and the concept of triangular arbitrage to exploit discrepancies in cross-rates.

Key Concepts:

Cross-Rate Calculation

Cross-rates are calculated by using two exchange rates involving three different currencies to derive the exchange rate between the two currencies that do not have a direct exchange rate quoted. For example, if you have rates for USD/EUR and CAD/USD, you can calculate the rate for CAD/EUR.

Market Conventions

FX market conventions include quoting exchange rates in a specific order (e.g., USD/EUR) and using defined conventional currency pairs. The market also avoids using direct or indirect quotes, which can vary based on location.

Inversion of Exchange Rates

Sometimes, to calculate a cross-rate, it is necessary to invert an exchange rate to align the currencies properly for multiplication. This is often done when the direct quote is not available for the desired cross-rate calculation.

Triangular Arbitrage

This involves exploiting the price discrepancies in cross-rates among three currencies to make a risk-free profit. It continues until the market corrects the mispricing, aligning with the efficient market hypothesis.

Formulas:

Cross-Rate Formula

CADEUR=CADUSD×USDEURCAD_{EUR} = CAD_{USD} \times USD_{EUR}

To calculate the cross-rate between CAD and EUR, multiply the exchange rate from CAD to USD with the exchange rate from USD to EUR.

Variables:
CADEURCAD_{EUR}:
Cross-rate between Canadian Dollar and Euro
CADUSDCAD_{USD}:
Exchange rate between Canadian Dollar and US Dollar
USDEURUSD_{EUR}:
Exchange rate between US Dollar and Euro
Units: currency units

Inverted Cross-Rate Formula

JPYCAD=(CADUSD)−1×JPYUSDJPY_{CAD} = (CAD_{USD})^{-1} \times JPY_{USD}

To calculate the cross-rate between JPY and CAD, first invert the exchange rate from CAD to USD, then multiply it by the exchange rate from USD to JPY.

Variables:
JPYCADJPY_{CAD}:
Cross-rate between Japanese Yen and Canadian Dollar
CADUSDCAD_{USD}:
Exchange rate between Canadian Dollar and US Dollar
JPYUSDJPY_{USD}:
Exchange rate between Japanese Yen and US Dollar
Units: currency units

Forward Rate Calculations

Learning Outcome Statement:

explain the arbitrage relationship between spot and forward exchange rates and interest rates, calculate a forward rate using points or in percentage terms, and interpret a forward discount or premium

Summary:

This LOS covers the relationship between spot and forward exchange rates, interest rates, and how these are influenced by arbitrage opportunities. It explains how forward rates are calculated using points or percentages and how to interpret forward discounts and premiums. The content also delves into the arbitrage relationships that ensure no riskless profit can be made due to discrepancies in these rates, and how forward rates can be used as a hedging tool in financial markets.

Key Concepts:

Forward Rate Calculation

Forward rates are typically quoted in terms of points, which represent the difference between the forward and spot rates, scaled to the last decimal of the spot rate. These points indicate whether the base currency is at a forward premium or discount.

Arbitrage Relationship

The arbitrage relationship ensures that the returns from investing in different currencies for the same period at respective risk-free rates, after accounting for the forward rate, are equal. This prevents arbitrage opportunities and ensures market efficiency.

Forward Discounts and Premiums

A forward discount or premium indicates the expected depreciation or appreciation of the base currency relative to the price currency. This is influenced by the interest rate differential between the two currencies.

Formulas:

Forward Rate Calculation

Ff/d=Sf/d(1+rf1+rd)F_{f/d} = S_{f/d} \left( \frac{1 + r_f}{1 + r_d} \right)

This formula calculates the forward rate based on the spot rate and the interest rates in the domestic and foreign countries, ensuring no arbitrage opportunity exists.

Variables:
Ff/dF_{f/d}:
forward exchange rate (foreign/domestic)
Sf/dS_{f/d}:
spot exchange rate (foreign/domestic)
rfr_f:
foreign risk-free interest rate
rdr_d:
domestic risk-free interest rate
Units: currency units

Percentage Forward Points Calculation

Ff/d−Sf/d=Sf/d(rf−rd1+rdτ)τF_{f/d} - S_{f/d} = S_{f/d} \left( \frac{r_f - r_d}{1 + r_d \tau} \right) \tau

This formula shows how the forward points, which are the difference between the forward and spot rates, depend on the interest rate differential and the term of the forward contract.

Variables:
Ff/dF_{f/d}:
forward exchange rate (foreign/domestic)
Sf/dS_{f/d}:
spot exchange rate (foreign/domestic)
rfr_f:
foreign risk-free interest rate
rdr_d:
domestic risk-free interest rate
τ\tau:
time fraction of the interest rate period
Units: currency units
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