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Topics

• Introduction
• Cross-Rate Calculations
• Forward Rate Calculations
  • Arbitrage Relationships
  • Forward Discounts and Premiums

Table of Contents

Introduction

• How to calculate cross exchange rates
• How to calculate forward exchange rates

Cross-Rate Calculations

Given two exchange rates and three currencies, it is possible to determine the third exchange rate. This way of determining the third exchange rate by converting one foreign exchange quote into another is called the cross rate.

Given the two exchange rates below, what is the PKR/INR rate?

PKR/INR= (PKR/USD) * (USD/INR)

Triangular arbitrage: If the implied cross rate is not equal to the quoted cross rate, then an arbitrage opportunity exists and it is called triangular arbitrage. In such cases, one would profit by buying low and selling high.

Forward Rate Calculations

Spot Exchange rates: Spot transactions involve the exchange of currencies for immediate delivery. For most currencies, this corresponds to “T+2” delivery, meaning that the exchange of currencies is settled two business days after the trade is agreed to by the two sides of the deal. The exchange rate used for such transactions is called the spot exchange rate.

Forward Exchange rates: Forward contracts are agreements to exchange one currency for another on a future date at an exchange rate agreed upon today. Any exchange transaction that has a settlement date later than T+2 is a forward contract. The exchange rate used for such transactions is called the forward exchange rate.

Forward exchange rates are generally quoted in terms of points or pips. The table below lists the spot rate and several forward rates for USD/EUR currency pair. The negative sign indicates that the forward rate is less than the spot rate, i.e., the base currency EUR is weakening relative to the US dollar.

Relationship between Spot Rates, Forward Rates, and Interest Rates

The forward exchange rate depends on relative interest rates. To derive the relationship between spot rates, forward rates, and interest rates, assume you have one unit of domestic currency to be invested for let us say, one year.

There are two options you may consider:

Option 1: Invest one unit of domestic currency (base currency) at the domestic risk-free rate $i_B$ for one period. Amount at the end of the period = $1+i_B$

Option 2: Convert one unit of domestic currency into foreign currency (base currency) today using the spot rate $S_{P/B}$. Invest this amount at the foreign risk-free rate $i_P$ for one period. Determine the forward rate $F_{P/B}$ today at which the price currency will be converted back to base currency.

Amount of units of price currency at the end of the period = ($S_{P/B}$) ($1+i_P$)
Amount in terms of base (domestic currency) at the end of the period = ($S_{P/B}$) ($1+i_P$) ($\frac{1}{F_{P/B}}$)

Since the risk of these two investments is the same, they should generate equivalent returns. As an investor, you should be indifferent between the two as it is a no-arbitrage relationship.

where:

• $F_{P/B}$ = forward rate
• $S_{P/B}$ = spot rate
• $i_P$ = risk-free rate of the price currency
• $i_B$ =risk-free rate of the base currency

Some important points to be noted:

• The currency with the higher (lower) interest rate will always trade at a discount (premium) in the forward market.

• This relationship ensures that there is no arbitrage. The only forward rate that prevents arbitrage is 108.91. Otherwise, for any rate greater/lesser than 108.91 traders can exploit the mispricing by buying low and selling high.
• If the forward contract is for x days, make an adjustment based on the x/360 convention unless told otherwise.