3.7: Exchange Rates and Currency Exchange

Introduction

You have set aside [latex]\$6,000[/latex] in Canadian funds toward hostel costs during a long backpacking trip through the United States, Mexico, and Europe. After searching on the Internet, you decide to use Hotwire.com to reserve your hostel rooms. The website quotes you the following amounts for each country: [latex]1,980[/latex] US dollars, [latex]21,675[/latex] Mexican pesos, and [latex]1,400[/latex] euros. Have you allocated enough money to cover these costs?

Whether you are a consumer backpacking around the world on distant vacations, investing in international securities, or shopping online at global Internet sites, you must pay for your purchases in local currencies out of your Canadian currency accounts. Businesses are no different as they export and import products to and from other countries. With outsourcing on the rise, it is also common for business services such as call centres and advertising agencies to be located abroad along with manufacturing facilities. Large-scale companies may have operations in several countries throughout the world.

All of these transactions and operations require the conversion of Canadian currency to a foreign currency or vice versa. This section shows you the basics of currency conversion rates and then explores finer details such as charges for currency conversion and what happens when one currency gets stronger or weaker relative to another.

Exchange Rates

An exchange rate between two currencies is defined as the number of units of a foreign currency that are bought with one unit of the domestic currency, or vice versa. Since two currencies are involved in every transaction, two published exchange rates are available. Let’s use Canada and the United States to illustrate this concept.

• The first exchange rate indicates what one dollar of Canada’s currency becomes in US currency.
• The second exchange rate indicates what one dollar of US currency becomes in Canadian currency.

These two exchange rates allow Canadians to determine how many US dollars their money can buy and vice versa. These currency rates have an inverse relationship to one another: if [latex]1[/latex] Canadian dollar equals [latex]0.80[/latex] US dollars, then [latex]1[/latex] US dollar equals [latex]\frac{1}{\$0.80}=1.25[/latex] Canadian dollars.

Most Canadian daily and business newspapers publish exchange rates in their financial sections. Although exchange rates are published in a variety of ways, a currency cross-rate table, like the table below, is most common. Note that exchange rates fluctuate all of the time as currencies are actively traded in exchange markets. Therefore, any published table needs to indicate the date and time at which the rates were determined. Also note that the cells where the same currency appears show no published rate as you never need to convert from Canadian dollars to Canadian dollars!

In the table, all exchange rates have been rounded to four decimals. In true exchange markets, most exchange rates are expressed in [latex]10[/latex] decimals or more such that currency exchanges in larger denominations are precisely performed. For the purposes of this textbook, we will use a four decimal standard to simplify the calculations while still illustrating the principles of currency exchange.

Technically, only half of the table is needed, since one side of the diagonal line is nothing more than the inverse of the other. For example, the euro is [latex]0.7137[/latex] per C$ on the bottom of the diagonal. The inverse, or [latex]\frac{1}{0.7137}=1.4011[/latex] per €, is what is listed on top of the diagonal (the difference of [latex]0.0001[/latex] is due to rounding to four decimals).

The formula is yet again another adaptation of Formula 3.1b[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex] on Rate, Portion, and Base. Formula 3.7 expresses this relationship in the language of currency exchange.

[latex]\boxed{3.7}[/latex] Currency Exchange 

[latex]\color{red}{\text{Desired Currency}}\color{black}{\text{ is what you are converting to:}}[/latex] In Formula 3.1b[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex], the desired currency represents the portion. This is what your money is worth in the other currency.

[latex]\color{blue}{\text{Exchange Rate}}\color{black}{\text{ is per unit of Current Currency:}}[/latex] In Formula 3.1b[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex], the exchange rate represents the rate, which is the current exchange rate per currency unit in which your money is currently valued. Looking at a Cross-Rate table, find the column that represents your existing currency and then cross-tabulate this column with the row for the desired currency (so you have desired currency per current currency). For example, if you have one euro that you want to convert into yen, locate the euro column and the yen row. The exchange rate is [latex]131.5789[/latex]. It is critical that both the exchange rate and current currency always be expressed in the same unit of currency type.

[latex]\color{green}{\text{Current Currency}}\color{black}{\text{ is what you are converting from:}}[/latex] In Formula 3.1b[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex], the current currency represents the base, which is the amount of money (number of units) in your existing currency.

This section opened with your backpacking vacation to the United States, Mexico, and Europe, for which you were quoted prices of [latex]\text{US}\$1,980[/latex], [latex]\text{MXN}\$21,675[/latex], and [latex]\text{€}1,400[/latex] for hostels. Assume all purchases are made with your credit card and that your credit card company charges [latex]2.5\%[/latex] on all currency exchanges. Can your [latex]\text{C}\$6,000[/latex] budget cover these costs?

Step 1: There are three currency amounts: [latex]\text{US}\$1,980[/latex], [latex]\text{MXN}\$21,675[/latex], and [latex]\text{€}1,400[/latex]. Using the cross-rate table, the Canadian exchange mid-rate per unit of each of these currencies is [latex]\text{US}\$0.9787[/latex], [latex]\text{MXN}\$0.0823[/latex], and [latex]\text{€}1.4012[/latex].

Step 2: Calculate the buy rates (since you are converting foreign currency into domestic currency) for each currency:

[latex]\text{US}\$=0.9787(1.025)=1.0032[/latex]

[latex]\text{MXN}\$=0.0823(1.025)=0.0844[/latex]

[latex]\text{€}=1.4012(1.025)= 1.4362[/latex]

Step 3: Apply Formula 3.7
([latex]\text{Desired Currency}=\text{Exchange Rate}\times\text{Current Currency}[/latex])
to each of these currencies:

[latex]\text{US\$: Desired Currency}=\text{US}\$1,980\times 1.0032=\text{C}\$1,986.34[/latex]

[latex]\text{MXN\$: Desired Currency}=\text{MXN}\$21,675\times 0.0844=\text{C}\$1,829.37[/latex]

[latex]\text{€: Desired Currency}=\text{€}1,400\times 1.4362=\text{C}\$2,010.68[/latex]

Putting the three amounts together, your total hostel bill is:

[latex]\$1,986.34+\$1,829.37+\$2,010.68=\$5,826.39[/latex]

Because this is under budget by [latex]\$173.61[/latex], all is well with your vacation plans.

Paths To Success

Let the buyer beware when it comes to international transactions. If you have ever purchased and returned an item to an international seller, you may have noticed that you did not receive all of your money back. For most consumers, international purchases are made via credit cards. What most consumers do not know is that the credit card companies do in fact use buy and sell rates that typically charge [latex]2.5\%[/latex] of the exchange rate when both buying and selling.

For example, if you purchase a [latex]\text{US}\$2,000[/latex] item at the rates listed in the cross-rate table, your credit card is charged [latex]$2,000\times 0.9787(1.025)=\$2,006.40[/latex]. If you return an item you do not want, your credit card is refunded [latex]\$2,000\times 0.9787(0.975)=\$1,908.40[/latex]. In other words, you are out [latex]\$2,006.40−\$1,908.40=\$98[/latex]! This amount represents your credit card company’s charge for the currency conversion—a whopping [latex]4.9\%[/latex] of your purchase price!

Currency Appreciation and Depreciation

An analyst on Global News is discussing how the Canadian dollar has strengthened against the US dollar. Your first reaction is that a strong Canadian dollar ought to be good thing, so hearing that this change might hurt Canada’s exports, you wonder how that could be.

Currencies are actively traded in the international marketplace, which means that exchange rates are changing all the time. As such, exchange rates rise and decline. A currency appreciates (or strengthens) relative to another currency when it is able to purchase more of that other currency than it could previously. A currency depreciates (or weakens) relative to another currency when it is able to purchase less of that other currency than it could previously. Take a look at two examples illustrating these concepts:

• EXAMPLE 1: If [latex]\text{C}\$1[/latex] buys [latex]\text{US}\$1.02[/latex] and the exchange rate rises to [latex]\text{US}\$1.03[/latex], then your [latex]\text{C}\$1[/latex] purchases an additional penny of the US dollars. Therefore, the Canadian currency appreciates, or strengthens, relative to the US dollar.
• EXAMPLE 2: Similarly, if the exchange rate drops to [latex]\text{US}\$1.01[/latex], then your [latex]\text{C}\$1[/latex] purchases one less penny of the US dollars. Therefore, the Canadian currency depreciates, or weakens, relative to the US dollar.

These concepts are particularly important to international business and global economies. Generally speaking, when a currency appreciates it has a positive effect on imports from the other country because it costs less money than it used to for domestic companies to purchase the same amount of products from the other country. However, the currency appreciation tends to also have a negative effect on exports to other countries because it costs the foreign companies more money to purchase the same amount of products from the domestic companies.

Key Takeaways

It is critical to observe that if currency A appreciates relative to currency B, then the opposite is true for currency B relative to currency A (currency B depreciates relative to currency A). The figure to the right illustrates this concept. Recalling Example 1 above, the Canadian currency appreciated, so the US currency depreciated. In Example 2, the Canadian currency depreciated, so the US currency appreciated.

Things To Watch Out For

It is very easy to confuse the two relative currencies, their values, and the concepts of appreciation and depreciation. For example, if the exchange rate increases between currency A relative to a unit of currency B, which exchange rate appreciated? If currency A has increased per unit of currency B, then it takes more money of currency A to buy one unit of currency B. As a result, currency B appreciates because a single unit of currency B can now buy more of currency A. For example, if the exchange rate is [latex]\text{US}\$1.0218[/latex] relative to [latex]\text{C}\$1[/latex] and it increases to [latex]\text{US}\$1.0318[/latex] relative to [latex]\text{C}\$1[/latex], then the Canadian dollar purchases one penny more. The figure helps you understand the relationships involved and provides a visual reminder of which direction everything is moving.

 

Section 3.7 Exercises

[latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex]THE FOLLOWING LATEX CODE IS FOR FORMULA TOOLTIP ACCESSIBILITY. NEITHER THE CODE NOR THIS MESSAGE WILL DISPLAY IN BROWSER.[latex]\text{Desired Currency}=\text{Exchange Rate}\times\text{Current Currency}[/latex][latex]\begin{align*}\text{Rate}=\frac{\text{Portion}}{\text{Base}}\end{align*}[/latex][latex]\text{S}=\text{C}+\text{E}+\text{P}[/latex]

Attribution

“7.3: Exchange Rates and Currency Exchange” from Business Math: A Step-by-Step Handbook (2021B) by J. Olivier and Lyryx Learning Inc. through a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License unless otherwise noted.