Written: Jun 4, 2005 Updated: Oct 20–26, 2008 (added section on realized and unrealized gains) Updated: Nov 15, 2010 (changed sectioning) Updated: Jan 2, 2011 (added Australia info to Section 5.3) Updated: Jun 2, 2015 (fixed typo in Table 5.2, reported by Alex Chiang) Updated: Aug 14, 2015 (fixed typo in Table 3.2, reported by Francisco Milan Campos) Updated: Nov 21, 2021 (fixed order of columns in Table 4.5)

Contents

• 0. Introduction

• 1. A brief review of double-entry accounting

• 1.1. Single-entry accounting

• 1.2. Net worth and equity

• 1.3. Double entry accounting

• 1.4. Income and Expenses

• 1.5. The Fundamental Accounting Equation

• 2. Foreign currency accounting issues

• 2.1. Single-entry foreign currency accounting

• 2.2. Double entry foreign currency accounting, the wrong way

• 2.3. Foreign exchange gains and losses

• 2.4. Calculating gains and losses through adjustments

• 3. Foreign currency accounting according to SSAP 20

• 3.1. Accounting for non-monetary current assets

• 3.2. The "official" solution: translation to a single currency

• 4. Foreign currency accounting using currency trading accounts

• 4.1. Restoring symmetry to foreign currency accounting

• 4.2. Accounting with currency trading accounts

• 4.3. Equivalence with SSAP 20

• 4.4. Example: tracking exchange gains and losses by source

• 5. Realized and unrealized gains and losses

• 5.1. An example from real estate

• 5.2. Importance of the "identity" of assets

• 5.3. The FIFO, LIFO, and adjusted cost base methods

• 6. Translation issues

• 6.1. Independence of viewpoint

• 6.2. Reference currency vs. Equity currency

• 7. Specific software

• 7.1. GnuCash

0. Introduction

Accounting with multiple currencies can be tricky. It is easy enough to use an exchange rate to translate between different currencies. But things get less obvious when one takes into account how exchange rates fluctuate over time. Multi-currency transactions can incur foreign exchange gains or losses. How does one account for these, while respecting the Fundamental Accounting Equation? Do different accounting methods give different results? What are their advantages and disadvantages? Does one need to pick a single reference currency, and why? And most importantly, how does all of this work?

As a non-expert, I found it surprisingly difficult to find useful information on this subject. I searched online, and I also consulted about fifteen different accounting textbooks in the U.K. and Canada. Of the few books that mentioned foreign currencies at all, most contained only a single paragraph on the topic, stating that foreign currency transactions should be translated to the local currency. I also studied the foreign currency support in the accounting software I was using (GnuCash), and I quickly found that the software did not balance my accounts correctly when multiple currencies were involved.

After much frustration and some additional thought and research, I believe I arrived at an understanding how multiple currency accounting should work, and why. So I wrote this document as a resource for those who might be looking for such information. The information in this tutorial is a synthesis of what I learned from books, what I learned from accounting standards such as SSAP 20, and what I figured out by myself. Since I am not an accounting expert, I am sure that some incorrect terminology has slipped into this document. If you find a mistake, I would be happy if you send me a correction. Aside from terminology issues, I am reasonably certain that the concepts described in this tutorial are sound.

Disclaimer: I am not an accountant, and I am not qualified to offer any kind of accounting advice. This tutorial expresses my opinion on how accounting with multiple currencies should work. I do not claim anything about the accuracy or legality of any of the information presented here, nor do I warrant its correctness or applicability in any way. Although I cite some accounting standards documents, I do not claim that this tutorial conforms with accepted accounting standards. If you need accounting advice, please consult a professional accountant.

1. A brief review of double-entry accounting

1.1. Single-entry accounting

Users of accounting software fall into two main classes: home users and business users. Most home users are only interested in tracking assets and liabilities. Assets are what you own (such as money in your bank account or cash in your pocket), and liabilities are what you owe (such as the balance on your credit card account, or money you owe your neighbor). An account keeps a running total of one particular type of asset or liability. A transaction is an increase or decrease of the balance of one or more accounts.

Table 1.1: Single-entry personal accounting

Date Description Bank Account (asset) Cash (asset) Credit Card (liability) Jan 1 Opening balance $1,000 $20 $600 Jan 4 Get paid + $200 Balance $1,200 $20 $600 Jan 5 Buy food with credit card + $70 Balance $1,200 $20 $670 Jan 12 Withdraw cash − $100 + $100 Balance $1,100 $120 $670 Jan 15 Pay credit card bill − $670 − $670 Balance $430 $120 $0 Jan 18 Get paid + $200 Balance $630 $120 $0 Jan 20 Buy food with cash − $65 Balance $630 $55 $0 Jan 23 Buy book with cash − $16 Balance $630 $39 $0

Table 1.1 shows an example of single-entry personal accounting with three accounts. It uses just a single currency, the Canadian Dollar. Note that account balances are shown on yellow lines, and transactions are shown on white lines. Also note that in asset accounts, positive numbers mean "more assets", whereas in liability accounts, positive numbers mean "more liability". Thus, transferring money from one asset account to another (as on January 12) leads to an increase in one account and a decrease in the other. But transferring money from an asset account to a liability account (as in paying a credit card bill on January 15) leads to a decrease in both accounts.

1.2. Net worth and equity

A person’s net worth is equal to their total assets minus their total liabilities. In other words, the net worth is the effective amount of money (and property) they own after compensating for all outstanding debts. A person’s net worth can be negative if the total liabilities exceed the total assets (this is called "living above one’s means", and is not recommended). In a business context, net worth is also called equity. We therefore have the first accounting equation: Assets − Liabilities = Equity.

Going back to the example from Table 1.1, note that some transactions, like "get paid" on January 4 and 18, increase a person’s net worth. This is called income. Other transactions, like "buy food" or "buy books" on January 5, 20, and 23, decrease a person’s net worth. This is called an expense. Still other transactions, such as "withdraw cash" on January 12 or "pay credit card bill" on January 15, do not affect the net worth at all.

The following chart shows the same transactions as Table 1.1, but this time we also show the net worth in a separate column.

Table 1.2: Personal accounting with net worth shown

Date Description Bank Account (asset) Cash (asset) Credit Card (liability) Net Worth (equity) Jan 1 Opening balance $1,000 $20 $600 $420 Jan 4 Get paid + $200 + $200 Balance $1,200 $20 $600 $620 Jan 5 Buy food with credit card + $70 − $70 Balance $1,200 $20 $670 $550 Jan 12 Withdraw cash − $100 + $100 Balance $1,100 $120 $670 $550 Jan 15 Pay credit card bill − $670 − $670 Balance $430 $120 $0 $550 Jan 18 Get paid + $200 + $200 Balance $630 $120 $0 $750 Jan 20 Buy food with cash − $65 − $65 Balance $630 $55 $0 $685 Jan 23 Buy book with cash − $16 − $16 Balance $630 $39 $0 $669

Note that the accounting equation "Assets − Liabilities = Equity" is not only satisfied by the balances (yellow lines), but also by the individual transactions (white lines).

1.3. Double entry accounting

Let us promote the equity column from Table 1.2 by calling it an "account". Then Table 1.2 is a simple instance of double entry accounting. The idea is to keep a separate running total for equity, in addition to the running totals for assets and liabilities. Each transaction must be balanced, which means, it must satisfy the accounting equation.

Historically (i.e., before computers were used), one of the purposes of double entry accounting was to catch arithmetic errors and data entry errors. Since each transaction had to be recorded in at least two places, the results of additions and subtractions could be double-checked simply by checking whether the accounts were balanced.

Today, arithmetic mistakes are no longer a major issue (although data entry errors still are). However, there is one other important benefit of double-entry accounting. Namely, it gives a convenient way of tracking and classifying income and expenses.

1.4. Income and Expenses

For businesses, it is important to keep track not just of assets and liabilities, but also of income and expenses. For example, a business needs to classify its income according to source. This is necessary both for tax reasons (income from different sources may be taxed differently), and also for business reasons (it helps to determine which parts of the business are profitable).

In the asset/liability method of tracking finances, money is classified according to its location: how much money is in your bank account, and how much in your wallet? By contrast, in tracking income and expenses, one must classify money according to its purpose: how much money did you spend on food, and how much on books?

For example, in Table 1.2, the total money spent on food in the given time period was $135 (namely, $70 on January 5 and $65 on January 20). Some of the food was bought with cash, and some with a credit card. While these different payment methods matter from the point of view of assets and liabilities, they are irrelevant from the point of view of expenses.

The most convenient way to track income and expenses is to split the column for "equity" from Table 1.2 into several columns. This is normally done for a fixed time period, for example, one year (called an "accounting period"). The equity at the beginning of the time period is called the "initial capital". All increases in equity since the beginning of the time period are called "income", and all decreases are called "expenses". We therefore have the second accounting equation: Equity = Capital + Income − Expenses.

We will therefore split the "equity" account into three columns called "initial capital", "income" and "expense". The income and expense columns can be further subdivided to account for different types of income and expenses. To keep our example simple, we will consider a single income account called "salary", and two expense accounts called "food" and "books".

Table 1.3: Personal accounting with income/expense tracking

Date Description Bank Account (asset) Cash (asset) Credit Card (liability) Initial Capital (capital) Salary (income) Food (expense) Books (expense) Jan 1 Opening balance $1,000 $20 $600 $420 $0 $0 $0 Jan 4 Get paid + $200 + $200 Balance $1,200 $20 $600 $420 $200 $0 $0 Jan 5 Buy food with credit card + $70 + $70 Balance $1,200 $20 $670 $420 $200 $70 $0 Jan 12 Withdraw cash − $100 + $100 Balance $1,100 $120 $670 $420 $200 $70 $0 Jan 15 Pay credit card bill − $670 − $670 Balance $430 $120 $0 $420 $200 $70 $0 Jan 18 Get paid + $200 + $200 Balance $630 $120 $0 $420 $400 $70 $0 Jan 20 Buy food with cash − $65 + $65 Balance $630 $55 $0 $420 $400 $135 $0 Jan 23 Buy book with cash − $16 + $16 Balance $630 $39 $0 $420 $400 $135 $16

As there is no limit to the number of accounts one can have, there is no limit on the number of different types of income and expenses that can be tracked using this system. Note that expenses (like liabilities) are recorded as positive numbers, even though they contribute to equity negatively.

Also note that the "initial capital" account never changes in this example. In business use, there are other types of capital besides "initial capital", and some of these other types are changeable. For example, if somebody invests money in a business (thereby becoming an owner), then this is considered an increase in capital rather than income. Thus, in general, there can be more than one capital account.

1.5. The Fundamental Accounting Equation

Putting together the first and second accounting equations Assets − Liabilities = Equity and Equity = Capital + Income − Expenses, we obtain the following, which is called the Fundamental Accounting Equation: Assets − Liabilities = Capital + Income − Expenses. The reader should check that both the yellow and the white rows in Table 1.3 indeed satisfy this equation.

Remark: "equity" is sometimes called "capital" and vice versa; I am not really sure of the most correct terminology. In this tutorial, I will continue to use these terms with their meanings introduced above.

Remark: Traditionally, accountants did not like to subtract or to use negative numbers. The accounting equation is therefore often written as "Assets + Expenses = Capital + Income + Liabilities". The quantities on the left-hand side of this last equation are sometimes called activa (Assets and Expenses), and those on the right-hand side are sometimes called passiva (Capital, Income, and Liabilities). The reader will easily verify that the two versions of the accounting equation are mathematically equivalent. Conceptually and pedagogically, I find the division into location (Assets and Liabilities) and purpose (Capital, Income, and Expenses) more useful than the division into activa and passiva. Therefore, in this tutorial, I will continue to use the version of the accounting equation that uses subtraction.

2. Foreign currency accounting issues

2.1. Single-entry foreign currency accounting

Handling foreign currencies in single-entry accounting is very simple. Each asset or liability account can have its own currency. When transferring money from one account to another, we simply use the exchange rate that is in effect at the time of the transfer.

Suppose you keep two kinds of cash in your pocket: Canadian dollars (CAD) and U.S. dollars (USD). On January 1, you have CAD 200 and no U.S. dollars. On January 2, you exchange CAD 120 for USD 100 at a bank (at the day’s exchange rate, which is 1.20 CAD/USD). On January 3, you spend USD 40 on food. On January 5, you exchange USD 60 for CAD 75 (at that day’s rate of 1.25 CAD/USD). On January 7, you spend CAD 20 on food. These transactions can easily be accounted for with two asset accounts:

Table 2.1: Single-entry foreign currency accounting

Date Description Canadian Cash (asset) U.S. Cash (asset) Jan 1 Opening balance CAD 200 USD 0 Jan 2 Exchange (1 USD = 1.20 CAD) − CAD 120 + USD 100 Balance CAD 80 USD 100 Jan 3 Buy food − USD 40 Balance CAD 80 USD 60 Jan 5 Exchange (1 USD = 1.25 CAD) + CAD 75 − USD 60 Balance CAD 155 USD 0 Jan 7 Buy food − CAD 20 Balance CAD 135 USD 0

As the example shows, foreign currencies are easy to account for as long as you only want to keep track of assets (and it is equally easy for liabilities). The process of doing currency transfers with exchange rates can easily be automated in software.

2.2. Double entry foreign currency accounting, the wrong way

Things get more complicated if we also want to keep track of income and expenses. To illustrate this, we follow the method from Table 1.3 and add accounts for initial capital and expenses. (Since there is no income in this example, we don’t need an income account). Because I live in Canada, we will keep the capital and expense accounts in Canadian dollars. Of course, when there is a transaction that affects one of these accounts, the amount will have to be translated to Canadian dollars. To do this, we have to know the exchange rate on January 3; let us assume for the sake of this example that this exchange rate was 1.30 CAD/USD, so that USD 40 = CAD 52 on that day.

Table 2.2: Double-entry foreign currency accounting, naively

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Food (expense) Jan 1 Opening balance CAD 200 USD 0 CAD 200 CAD 0 Jan 2 Exchange (1 USD = 1.20 CAD) − CAD 120 + USD 100 Balance CAD 80 USD 100 CAD 200 CAD 0 Jan 3 Buy food (1 USD = 1.30 CAD) − USD 40 + CAD 52 Balance CAD 80 USD 60 CAD 200 CAD 52 Jan 5 Exchange (1 USD = 1.25 CAD) + CAD 75 − USD 60 Balance CAD 155 USD 0 CAD 200 CAD 52 Jan 7 Buy food − CAD 20 + CAD 20 Balance CAD 135 USD 0 CAD 200 CAD 72

Note that we have recorded each transaction according to the exchange rate in effect on the date the transaction was made. Remember the accounting equation "Assets − Liabilities = Capital + Income − Expenses". Please convince yourself that each transaction (white line) indeed satisfies this equation, if one takes the daily exchange rate into account. Therefore, the white lines are balanced.

So what is the problem? The problem is that the yellow lines are not balanced. For example, consider the balance for January 7. We started with CAD 200, and we spent CAD 72 on food, so we should still have CAD 128 in our pocket. Yet, we have CAD 135. So something went wrong: we gained CAD 7 without any record of it on the "equity" side of the balance sheet. The accounting system of Table 2.2 violates the Fundamental Accounting Equation.

2.3. Foreign exchange gains and losses

The CAD 7 that were "won" in Table 2.2 are called a foreign exchange gain. Such gains are caused by exchange rate fluctuations between transactions. For example, if you buy a foreign currency, then sell it later at a higher price, you have enjoyed a foreign exchange gain. Similarly, if you lose money due to exchange rate fluctuations, then this is called a foreign exchange loss. For accounting purposes, foreign exchange gains count as income, and foreign exchange losses count as an expense. What the accounting method from Table 2.2 lacks is a way to track foreign exchange gains and losses.

The question we need to address is therefore how to calculate foreign exchange gains and losses, and how to record them, preferably while keeping the Fundamental Accounting Equation intact.

Calculating foreign exchange gains and losses is somewhat tricky, because such gains and losses do not occur during transactions, but rather in the time intervals between transactions. It is therefore not immediately clear at what time they should be entered in the balance sheet. In principle, gains and losses happen every time the exchange rate changes, whether or not there was a transaction. But it is clearly not feasible to record them on a continuous basis (for example, once per second). So there must be a better way.

Another potential complication is that the concept of a gain or loss depends on the "base" currency, and therefore on your point of view. Suppose you have CAD 150 and USD 100, and the exchange rate changes from 1.20 CAD/USD to 1.25 CAD/USD. If you are Canadian, then you have just enjoyed a gain (your equity increased from CAD 270 to CAD 275). But if you are American, then you have suffered a loss (your equity decreased from USD 225 to USD 220). This dependency on a base currency seems necessary, but it is also frustrating, as it seems to imply that a multi-currency accounting system cannot be "symmetric" in the way different currencies are treated. (It is in fact possible to design a symmetric system, as we will see in Section 6.1 below).

2.4. Calculating gains and losses through adjustments

In principle, one could account for foreign exchange gains and losses in the same way one accounts for intractable arithmetic errors: by adding an "adjusting entry" before each yellow line in Table 2.2, forcing the balances to obey the accounting equation. This method is shown in Table 2.3.

Table 2.3: Foreign currency accounting with adjustments

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Food (expense) Exchange gain (income) Jan 1 Opening balance CAD 200 USD 0 CAD 200 CAD 0 CAD 0 Jan 2 Exchange (1 USD = 1.20 CAD) − CAD 120 + USD 100 Balance CAD 80 USD 100 CAD 200 CAD 0 CAD 0 Jan 3 Buy food (1 USD = 1.30 CAD) − USD 40 + CAD 52 Adjustment + CAD 10 Balance CAD 80 USD 60 CAD 200 CAD 52 CAD 10 Jan 5 Exchange (1 USD = 1.25 CAD) + CAD 75 − USD 60 Adjustment − CAD 3 Balance CAD 155 USD 0 CAD 200 CAD 52 CAD 7 Jan 7 Buy food − CAD 20 + CAD 20 Balance CAD 135 USD 0 CAD 200 CAD 72 CAD 7

To calculate Table 2.3, we simply had to look at each day’s balance (yellow line), and force it to satisfy the accounting equation according to that day’s exchange rate. An adjusting entry was then added to produce the desired numbers. As a result, the "yellow" (balance) lines are balanced, but the "white" (transaction) lines are not.

Note that there was an exchange loss of CAD 3 on January 5. While this technically is an "expense", we have not created a special "expense" account for it; instead we have recorded this loss as a "negative gain". This is in accordance with an accepted accounting principle that states that exchange gains and losses can be directly offset against each other.

The method shown in Table 2.3 is undesirable for several reasons. The main problem is that the additional "adjusting entries" are themselves not balanced, thus violating the principle of double entry accounting. This removes a source of redundancy. In effect, since one constantly has to "force" the accounts to balance, one can no longer notice the difference between currency fluctuations and random arithmetical errors.

Moreover, even if the accounts are in balance at one point, they continually "go out of balance" as the exchange rate changes, even while no transactions occur. Thus, adjustment entries should really be added continuously, rather than once per transaction. This would be an undesirable thing to have to do.

Another problem with the adjustment method is that it makes it difficult to add or change transactions retroactively. Suppose we want to enter another transaction on January 2. Then all future adjusting balances (on January 3 and 5) have to be re-calculated. While it would be possible for a computer to do this automatically, it is not very elegant and even seems manipulative. Instead of simply recording transactions, we are now calculating transactions, and the separation of data entry and analysis seems to be violated. Cause and effect of individual transactions are difficult to track.

3. Foreign currency accounting according to SSAP 20

3.1. Accounting for non-monetary current assets

So if the methods shown in Tables 2.2 and 2.3 are bad, then how should double-entry accounting for multiple currencies be done? Before we look at possible solutions, it is useful to recall how one accounts for the changing value of non-monetary assets.

Imagine that you are in the business of buying and selling oil. Let us assume that oil is a current asset, which means that it can be easily converted to cash on the market. Therefore, oil should always be accounted for at its current market price. (This is different from non-current assets, such as real-estate, which are usually accounted for at their historical cost).

Suppose you buy oil at market prices, and you sell it at market price plus a profit margin of CAD 5 per barrel. On January 1, you buy 10 barrels of oil at a price of CAD 30 per barrel. On January 2, you sell 2 barrels for CAD 35 each (the market price is still CAD 30). On January 3, the price of oil increases to CAD 32 per barrel. On January 4, you sell 5 more barrels at CAD 37. We can account for this as follows:

Table 3.1: Accounting for non-monetary current assets

Date Description Cash (asset) Oil (asset) Inventory Initial capital (capital) Sales profit (income) Capital gain (income) Jan 1 Opening balance CAD 500 CAD 0 0 barrels CAD 500 CAD 0 CAD 0 Jan 1 Buy 10 barrels − CAD 300 + CAD 300 +10 barrels (@CAD 30) Balance CAD 200 CAD 300 10 barrels (@CAD 30) CAD 500 CAD 0 CAD 0 Jan 2 Sell 2 barrels + CAD 70 − CAD 60 −2 barrels (@CAD 30) + CAD 10 Balance CAD 270 CAD 240 8 barrels (@CAD 30) CAD 500 CAD 10 CAD 0 Jan 3 Re-value inventory + CAD 16 8 barrels (+CAD 2) + CAD 16 Balance CAD 270 CAD 256 8 barrels (@CAD 32) CAD 500 CAD 10 CAD 16 Jan 4 Sell 5 barrels + CAD 185 − CAD 160 −5 barrels (@CAD 32) + CAD 25 Balance CAD 455 CAD 96 3 barrels (@CAD 32) CAD 500 CAD 35 CAD 16

There are several things to note in this example. The first remark is that the green column marked "Inventory" is not an account; it is simply an informational column recording the number of barrels of oil we currently own, and their current price. This column does not usually appear in a company’s accounts; it can be kept separately. The accounts only contain the value of the assets, rather than the assets themselves.

Also note that this accounting method clearly distinguishes different sources of income: CAD 35 of the income is from sales profits (for 7 barrels sold at a profit margin of CAD 5 each), and CAD 16 is from capital gains (because the price of oil rose by CAD 2 at a time when you had 8 barrels).

From the viewpoint of multiple currency accounting, we remark that a "barrel of oil", while cumbersome to carry in your wallet, might nevertheless be regarded as a kind of currency. We make the following observations:

• All accounts are in CAD. Even though there is another currency involved, namely "barrels of oil", we do not actually keep any account in this currency. Our "oil asset" account is kept in CAD.
• Because any fluctuations in the price of oil are recorded in CAD, they are reflected on the asset side of the balance sheet and also on the income/expense side. As a result, the accounts are balanced at all times.
• Since the oil asset account is denominated in CAD and not in barrels, we must separately keep an inventory of the actual amount of oil owned, and also of the current market price. This inventory, which is shown in the green column in the example, is only informational; it is not part of the account sheet, and does not take part in any balancing.

3.2. The "official" solution: translation to a single currency

Most countries have a code of standard accounting practices that deals with foreign currency accounting. These can be searched for (and in some cases, found) on the web. Examples of such documents are SSAP 20 (for Canada and the U.K.) and FASB 52 (for the U.S.). It is my understanding that these various country-specific documents more or less all say the same thing. For example, SSAP 20 states:

During an accounting period, a company may enter into transactions which are denominated in a foreign currency. The result of each transaction should normally be translated into the company’s local currency using the exchange rate in operation on the date on which the transaction occurred [...].

In effect, this means that foreign currencies should be treated precisely in the same way as non-monetary assets, i.e., they should be accounted for in the local currency. (As before, we use Canadian dollars as the local currency in our examples). This means that, just as the oil account in Table 3.1 was denominated in CAD (and not in "barrels of oil"), the U.S. dollar account in our foreign currency example should also be denominated in CAD (and not in USD), when using this accounting method. Currency gains and losses are determined in the same way as other kinds of capital gains and losses: namely, by re-valuing accounts when a change of exchange rates occurs.

We illustrate this by reconsidering the example from Tables 2.1–2.3, using the SSAP 20 method. Note that the accounting principle used here is very similar to Table 3.1.

Table 3.2: Foreign currency accounting à la SSAP 20

Date Description CAD Cash (asset) USD Cash (asset) USD Cash Inventory Initial Capital (capital) Food (expense) Exchange gain (income) Jan 1 Opening balance CAD 200 CAD 0 USD 0 CAD 200 CAD 0 CAD 0 Jan 2 Exchange − CAD 120 + CAD 120 + USD 100 (@CAD 1.20) Balance CAD 80 CAD 120 USD 100 (@CAD 1.20) CAD 200 CAD 0 CAD 0 Jan 3 Re-value USD + CAD 10 USD 100 (+ CAD 0.10) + CAD 10 Balance CAD 80 CAD 130 USD 100 (@CAD 1.30) CAD 200 CAD 0 CAD 10 Buy food − CAD 52 − USD 40 (@CAD 1.30) + CAD 52 Balance CAD 80 CAD 78 USD 60 (@CAD 1.30) CAD 200 CAD 52 CAD 10 Jan 5 Re-value USD − CAD 3 USD 60 (− CAD 0.05) − CAD 3 Balance CAD 80 CAD 75 USD 60 (@CAD 1.25) CAD 200 CAD 52 CAD 7 Exchange + CAD 75 − CAD 75 − USD 60 (@CAD 1.25) Balance CAD 155 CAD 0 USD 0 CAD 200 CAD 52 CAD 7 Jan 7 Buy food − CAD 20 + CAD 20 Balance CAD 135 CAD 0 USD 0 CAD 200 CAD 72 CAD 7

Compare this carefully to Table 2.3. The first thing to notice is that each white and yellow row is balanced in Table 3.2, which was not the case in Table 2.3. On the equity side of the balance sheet (to the right of the solid black line), nothing has changed (except the order of some transactions). On the assets/liabilities side (to the left of the solid black line), the USD account is now denominated in CAD, and therefore, the currency fluctuations on January 3 and 5 lead to re-valuations of assets. As in Table 3.1, the green column is only informational and should be disregarded for balancing purposes.

We summarize the main points of the SSAP 20 method:

• There is always one special reference currency, typically "the company’s local currency".
• All accounts are kept in the reference currency.
• Tallies in non-reference currencies are only informational, and do not take part in balancing.
• Fluctuations in the relative values of non-reference currencies are reflected immediately in their corresponding asset accounts (and also in a currency gain/loss account).

The advantages of this method are:

• Since all entries are in one currency, no special software support for multiple currencies is needed — in principle, only single-currency accounting software is required.
• The accounts are balanced.

The disadvantages are:

• Separate "tallies" of foreign currencies and their exchange rates must be kept (the green columns in the examples). In fact, these "tallies" contain original (source) information, whereas the corresponding asset/liability accounts contain derived (calculated) information. In a sense, the foreign currency accounts that are kept in a local currency are only "virtual" accounts, whose contents are entirely calculated.
• Accounts must be re-valued periodically, and the question arises when to do this. In principle, this should happen just before every transaction, and just before generating a report. If there are many foreign currency accounts, this can mean a lot of (manual) work.
• It is relatively difficult to add or change a transaction retroactively. If a new transaction is entered for an earlier date, then all future currency gains and losses must be recalculated. This changes future transactions, not just future balances.
• There is an unsatisfying asymmetry in this accounting method, because one currency must be given prominence above all others.

The first three of these disadvantages can be alleviated by using software that supports the SSAP 20 accounting method. In this case, the user only has to enter the data from the green columns, and the software is able to take care of conversions, re-valuations, and calculating the corresponding gains and losses.

4. Foreign currency accounting using currency trading accounts

4.1. Restoring symmetry to foreign currency accounting

In Section 3, one of our main complaints about the SSAP 20 accounting method was that it introduces an apparent asymmetry by requiring one currency to be singled out as the "reference currency". We would prefer to work with accounts that are directly denominated in multiple currencies, but we have seen in Section 2 that this leads to problems with the accounting equation. Let us now investigate a method that restores symmetry while also respecting the accounting equation.

Consider the following simple scenario. On January 1, you have assets of CAD 60 and USD 100, and the exchange rate is 1.20 CAD/USD. Therefore, your total capital, in Canadian dollars, is CAD 60 + CAD 120 = CAD 180. Let us assume that you do not make any further transactions in January, but the exchange rate changes to 1.30 CAD/USD, 1.25 CAD/USD, and 1.15 CAD/USD on the three days following January 1. Recording only balances (yellow lines) and ignoring transactions (white lines) for the moment, we find that we should record the following, more or less in the style of Table 2.3:

Table 4.1: Cumulative exchange gain as a function of exchange rate

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Exchange gain (income) Jan 1 Balance (1 USD = 1.20 CAD) CAD 60 USD 100 CAD 180 CAD 0 Jan 2 Balance (1 USD = 1.30 CAD) CAD 60 USD 100 CAD 180 CAD 10 Jan 3 Balance (1 USD = 1.25 CAD) CAD 60 USD 100 CAD 180 CAD 5 Jan 4 Balance (1 USD = 1.15 CAD) CAD 60 USD 100 CAD 180 − CAD 5

Note that the only change from day to day is the exchange gain; there are no changes in the asset accounts or in the initial capital account. Also note that the accounts are balanced on each day, if one takes that day’s exchange rate into account.

Now let us save some work by giving a formula for the exchange gain, instead of calculating it explicitly each day. Let the exchange rate on a given day be x CAD/USD, and let y be the corresponding balance in the exchange gain account. What is the formula for y in terms of x? We must satisfy the Fundamental Accounting Equation "Assets − Liabilities = Capital + Income − Expense", and therefore we must have: CAD 60 + x * USD 100 = CAD 180 + y. This can easily be solved for y: y = x * USD 100 − CAD 120. In other words, the balance y of the exchange gain account on any given day is exactly equal to the current value of (USD 100 minus CAD 120), expressed in Canadian dollars. We can simplify Table 4.1 by expressing the exchange gain as a difference of multiple currencies, rather than expressing it as a value in a single currency.

Table 4.2: Cumulative exchange gain expressed as a difference of multiple currencies

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Exchange gain (income) Jan 1 Balance (1 USD = 1.20 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 2 Balance (1 USD = 1.30 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 3 Balance (1 USD = 1.25 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 4 Balance (1 USD = 1.15 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120

Note that the exchange gains shown in Table 4.2 are exactly identical to those shown in Table 4.1, given each day’s exchange rate. For example, on January 3, Table 4.2 shows (USD 100 − CAD 120), but this is equal to the amount of CAD 5 shown in Table 4.1, given that day’s exchange rate of 1.25 CAD/USD.

By expressing exchange gains in terms of multiple currencies, we have not altered the substance of what has been recorded. We have only changed the method of recording.

Note that each line of Table 4.2 is balanced, independently of the current exchange rate. Indeed, the entries of each currency are balanced separately: we have CAD 60 = CAD 180 − CAD 120, and also USD 100 = USD 100.

4.2. Accounting with currency trading accounts

An account that is denominated as a difference of multiple currencies, like the "exchange gain" account in Table 4.2, is known as a currency trading account. The following is an exact copy of Table 4.2, except that we have renamed the "exchange gain" account as "USD trading":

Table 4.3: Foreign currency accounting with a currency trading account

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) USD Trading (trading) Jan 1 Balance (1 USD = 1.20 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 2 Balance (1 USD = 1.30 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 3 Balance (1 USD = 1.25 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120 Jan 4 Balance (1 USD = 1.15 CAD) CAD 60 USD 100 CAD 180 USD 100 − CAD 120

It is important to note that a currency trading account does not represent assets. The purpose of a currency trading account is to calculate foreign exchange gains and losses. By convention, gains are recorded positively and losses negatively, and therefore a currency trading account is a kind of income account.

Currency trading accounts are different from any other account in that they are denominated in two (or more) currencies. Typically, one of these currencies is your "reference" currency, and the other one is in the usual terminology called a "security". Thus, in the above example, we may say that the USD trading account holds "securities" of USD 100, which are offset by CAD 120 in negative "currency".

A currency trading account reflects currency exchanges made in the past, and in an ideal situation where exchange rates never change, its balance is supposed to be zero. However, as the exchange rate fluctuates, so does the value of the currency trading account, and such fluctuations correspond exactly to currency gains and losses. At the end of an accounting period, one may move the actual ("realized") gain or loss to an income or expense account denominated in the reference currency.

Let us now illustrate the use of a currency trading account by redoing the example from Sections 2 and 3. We replace the "currency gain" account by a "currency trading" account between U.S. dollars and Canadian dollars.

Table 4.4: Foreign currency accounting with a currency trading account

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Food (expense) USD Trading (trading) Jan 1 Opening balance CAD 200 USD 0 CAD 200 CAD 0 USD 0 − CAD 0 Jan 2 Exchange (1 USD = 1.20 CAD) − CAD 120 + USD 100 + USD 100 − CAD 120 Balance CAD 80 USD 100 CAD 200 CAD 0 USD 100 − CAD 120 Jan 3 Buy food (1 USD = 1.30 CAD) − USD 40 + CAD 52 − USD 40 + CAD 52 Balance CAD 80 USD 60 CAD 200 CAD 52 USD 60 − CAD 68 Jan 5 Exchange (1 USD = 1.25 CAD) + CAD 75 − USD 60 − USD 60 + CAD 75 Balance CAD 155 USD 0 CAD 200 CAD 52 USD 0 + CAD 7 Jan 7 Buy food − CAD 20 + CAD 20 Balance CAD 135 USD 0 CAD 200 CAD 72 USD 0 + CAD 7

Note that transactions that do not involve currency conversions (like "buy food" on January 7) are entered in the usual way for double entry accounting. Transactions that do involve currency conversions (like "exchange" on January 2 and 5 or "buy food" on January 3) must go through the trading account. Conceptually, when exchanging Canadian dollars for U.S. dollars, we are moving Canadian dollars "into" the trading account, and an equivalent amount of U.S. dollars "out of" the trading account. However, there is a sign reversal, so money moved "into" the trading account is given a negative sign and money moved "out of" it is given a positive sign. [FIXME: is there a standard convention for signs on trading accounts?]

Comparing the use of a currency trading account (as in Table 4.4) to the other accounting methods shown in Tables 2.3 and 3.2, we note that the need for any periodic adjustments (as in Table 2.3, January 3 and 5) or for periodic re-valuation of assets (as in Table 3.2, January 3 and 5) has been eliminated. Indeed, the value of the trading account is "self-adjusting", because it is sensitive to exchange rates. Since no manual adjustments or re-valuations are needed, the question of when to record them becomes a moot point.

Another consequence of the self-adjusting nature of currency trading accounts is that there is no penalty for entering or changing transactions retroactively. Unlike in the SSAP 20 system, where retroactive transactions have an effect on future transactions and balances, in the currency trading account system, retroactive currency transactions only affect future balances (and not future transactions). In this respect, their effect is no different from other retroactive transactions.

Also note that, when using the currency trading account method as in Table 4.4, there is no need to keep a separate "tally" or "inventory" of foreign currency assets, as we had to do using the SSAP 20 method in Table 3.2 (green column). Instead, the foreign currency assets are held directly in an account.

Another advantage of the currency trading account method is that it provides some flexibility in tracking income and expenses from currency exchange. For many applications, it will be sufficient to have just a single currency trading account. But it is also possible to have several such accounts, for instance, one account for each foreign currency, or one account for each branch of your business, or one account for each customer. This makes it possible to keep track of foreign exchange gains and losses attributed to different sources.

There is no reason that a single currency trading account has to be limited to two currencies. It is possible to set up a currency trading account to account for gains and losses from trades between three or more currencies. The balance in such an account is expressed as a sum or difference of amounts from each of the participating currencies.

Here is a summary of the main advantages of using currency trading accounts for multi-currency accounting:

• Foreign currency transactions are entered in accounts that are denominated directly in the foreign currency. Thus, the account sheet contains source information.
• Accounts are balanced; all transactions and balances respect the accounting equation.
• Foreign exchange gains and losses do not need to be calculated on a continuous basis; instead, they can be calculated whenever convenient (e.g. once per accounting period).
• Transactions can easily be entered or changed retroactively, without causing any change in future transactions.
• Foreign exchange gains and losses from different sources can be tracked independently (see Section 4.4 for an example).
• There is no "reference" currency and thus no inherent asymmetry in this accounting method.

The most important conceptual facts to remember about currency trading accounts are the following:

• The purpose of a currency trading account is not to perform conversions, but to calculate gains and losses.
• The value of a currency trading account does not change during transactions, but in the time intervals between transactions.

4.3. Equivalence with SSAP 20

The accounting method using currency trading accounts is mathematically equivalent to methods such as SSAP 20. By this, I mean that it can be used to generate financial reports that are identical to those generated under SSAP 20. Indeed, suppose we were to generate a financial report from the data in Table 4.4 at the end of January 3. We have recorded the following balances at the end of January 3:

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Food (expense) USD Trading (trading) Jan 3 Balance CAD 80 USD 60 CAD 200 CAD 52 USD 60 − CAD 68

Using the exchange rate of 1.30 CAD/USD in effect on January 3, we translate all balances to CAD. Note that USD 60 = CAD 78, and therefore USD 60 − CAD 68 = CAD 10. Here is the "translated" balance sheet:

Date Description Canadian Cash (asset) U.S. Cash (asset) Initial Capital (capital) Food (expense) USD Trading (trading) Jan 3 Balance CAD 80 CAD 78 CAD 200 CAD 52 CAD 10

Keeping in mind that a currency trading account is a special kind of income account representing a foreign exchange gain, the translated balance sheet is exactly identical to the end-of-day balances for January 3 obtained from the SSAP 20 method that are shown in Table 3.2.

Therefore, after translation to the "reference" currency, both accounting methods yield identical data and generate identical financial reports.

4.4. Example: tracking exchange gains and losses by source

Suppose you are a Canadian company selling goods to U.S. customers. Your books are mostly kept in Canadian dollars, but you invoice your customers in USD. Your customers are given 10 days to pay after you send them an invoice, and some additional time may pass until you receive and deposit their USD cheque in your CAD bank account. Foreign exchange gains and losses will occur if the exchange rate changes between the invoice date and the date the payment is received and converted to CAD.

Suppose that, for some reason, you would like to track foreign exchange gains and losses on a per-customer basis (for example, you want to determine whether it would be worthwhile to tighten some customer’s payment terms). This can easily be done by using one currency trading account for each customer.

Table 4.5: One currency trading account per customer

Date Description Bank Account (asset) U.S. Accounts Receivable (asset) Income (income) Trading Customer 1 (trading) Trading Customer 2 (trading) Jan 1 Opening balance CAD 0 USD 0 CAD 0 USD 0 − CAD 0 USD 0 − CAD 0 Jan 2 Invoice Customer 1 (1 USD = 1.20 CAD) + USD 100 + CAD 120 + USD 100 − CAD 120 Balance CAD 0 USD 100 CAD 120 USD 100 − CAD 120 USD 0 − CAD 0 Jan 5 Invoice Customer 2 (1 USD = 1.30 CAD) + USD 200 + CAD 260 + USD 200 − CAD 260 Balance CAD 0 USD 300 CAD 380 USD 100 − CAD 120 USD 200 − CAD 260 Jan 7 Payment Customer 1 (1 USD = 1.25 CAD) + CAD 125 − USD 100 − USD 100 + CAD 125 Balance CAD 125 USD 200 CAD 380 USD 0 + CAD 5 USD 200 − CAD 260 Jan 20 Payment Customer 2 (1 USD = 1.15 CAD) + CAD 230 − USD 200 − USD 200 + CAD 230 Balance CAD 355 USD 0 CAD 380 USD 0 + CAD 5 USD 0 − CAD 30

The trading account balances on January 20 show that your interaction with customer 1 resulted in a foreign exchange gain of CAD 5, whereas your interaction with customer 2 resulted in a foreign exchange loss of CAD 30. The fact that a particular foreign currency interaction is "finished", i.e., there are no outstanding currency exchanges, is reflected in the fact that the USD balance in the corresponding currency trading account is 0.

The same method can be used to track foreign exchange gains and losses associated with any division of sources that you care to keep track of. For example, you might want to separate exchange gains and losses from different parts of your business, or from different countries (this works even when several countries share the same currency).

5. Realized and unrealized gains and losses

So far, I have discussed a method for calculating gains and losses from currency transactions (and by extension, from transactions with other types of variable-priced assets, such as barrels of oil or shares of stock). Such gains and losses are also called capital gains and losses. In practice, it is often important to distinguish between realized and unrealized gains and losses. One area where this distinction is relevant is taxation: usually, only realized gains are taxable. The exact determination of whether a particular gain or loss is realized or unrealized is often tricky, and the rules vary from country to country. In this section, I will briefly discuss these issues, and show how they can fit into the multi-currency accounting method.

In general terms, when you own a variable-priced asset, market fluctuations will cause the value of the asset to change over time. These changes of value are called unrealized gains and losses as long as you still own the item. When you sell the item, they turn into a realized gain or loss. While the basic principle is simple, the precise rules for when a gain/loss is realized depend on details of the tax code and accounting practices. In particular, it is not always obvious how to determine the realized gain or l