7. The Account

Most introductory accounting textbooks provide an incorrect expression of the Accounting Equation and then attempt to make this expression correct by obfuscating the very meaning of debit and credit. The obfuscation is accomplished with a set of instructions that are typically worded something like the following:

“The signs reverse on opposite sides of the equal sign [of the Accounting Equation].” (1)

More explicitly, the folly is expressed:

“Pacioli perceived that, having designed the T-account with two sides in order to reflect increases and decreases, he could add still another algebraic balancing feature by reversing the position in the account of the ‘increases’ and ‘decreases’ on the opposite sides of the equal sign [of the Accounting Equation].” (2)

In fact, Pacioli never designed this “reversing” feature and was not even aware of the Accounting Equation. The idea of reversing the signs of account balances is a much more recent invention needed to make the Accounting Equation work. It is misguided for a number of reasons:

1. The Accounting Equation is typically written wrong with the signs reversed across the equal sign and, as compensation, the bookkeeper is told that he must reverse the very meaning of double-entry bookkeeping to make the equation work. If we correct the equation, this “reversal” is not necessary and the real meaning of double-entry bookkeeping is preserved. The bookkeeping remains the same; the bookkeeper gets to go home without his eyes crossed.

2. The instructions are blatantly wrong – some accounts do not follow this rule.

3. Double-entry bookkeeping is essentially a record of changes in financial resources from a credit state to a debit state, the direction of the change has nothing to do with where the affected accounts lie in the accounting equation. In fact, double-entry had worked for hundreds of years before the Accounting Equation existed.

The account is a separate “book” of the general ledger that receives debit and credit entries. Each debit represents the depositing of financial resources into the account and each credit represents the withdrawal of financial resources from the account. This is the only rule that should guide the bookkeeper’s activity. If the transaction represents a flow of resources away from the account, the account should be credited. If the transaction represents a flow of resources into the account, the account should be debited.

The “cash” account and the “accumulated depreciation” account are on the same side of the Accounting Equation and yet their respective balances are typically increased in opposite directions. The “cash” account should always have a debit balance representing the surplus of deposits (debits) over withdrawals (credits). Its balance is therefore increased by debits. The “accumulated depreciation” account, on the other hand, should always have a credit balance and therefore be increased by credit entries.

As an instruction for the bookkeeper, the rule is worthless. Many accounts receive large amounts of both debits and credits and therefore the instruction does not really provide a rule for his guidance. Its only purpose is to make sense out of an algebraic equation that is nonsense. The Accounting Equation is typically expressed incorrectly as:

Assets = Liabilities + Equity

When it should be written:

Assets = -(Liabilities + Equity)

By telling the bookkeeper that the signs of the accounts change from one side of the equation to the other, accounting textbooks are able to make negatives turn into positives. By way of an awkward shell game, wrongfully credited to poor Luca Pacioli, modern accounting textbooks have made assets equal to their opposite.

(1) Welsh, Glenn A. and Anthony, Robert N., Fundamentals of Financial Accounting (Homewood, Illinois: Richard d. Irwin, Inc., 1974) p. 89.

(2) Welsh, Glenn A. and Anthony, Robert N., Fundamentals of Financial Accounting (Homewood, Illinois: Richard d. Irwin, Inc., 1974) p. 88.

The Semantics of Double-Entry Bookkeeping

The following post was written and contributed by Cornelis ("Kees") A. Los, PhD, Professor of Finance. Faculty of Management, The University of Lethbridge (Alberta, Canada).

Assets (A), Liabilities (L) and Equity (E) can each be debited and credited, but these traditional double-entry bookkeeping terms are used mirror-like in the traditional equation in order to maintain the balance equality.

These bookkeeping terms and their effect on the particular item in brackets (.), thanks to the T-accounts, are:

Asset debit (+) and asset credit (-)

Liability debit (-) and liability credit (+)

Equity debit (-) and equity credit (+)

For example, if I debit an asset by $50, the asset's value is increased by $50 and to maintain the balance I have to credit $50 to a liability (if I borrow to pay for the asset increase) or I have to credit equity by $50 (if, as an owner, I invest my own money in the asset), so that +$50 = +$50

Therefore, the accounting "problem" resides in the terms "debit" and "credit, which, in Accounting, have different meaning depending on which side of the "=" sign you are" and not in the accounting equation A=L+E. This should not come as a surprise, since every new student of double-entry bookkeeping confuses these terms "debit" and "credit," since they have different meaning, depending on which side of the accounting equation you are. That that is a "semantic problem" and "confusing," I agree with. Words with double meaning depending on which side you're on, provides an Orwellian taste. But that is not a mathematical or logical problem, since there is nothing wrong with the accounting equation or even with using liabilities or equity as negative assets. It is a semantic problem.

(Between cultures there are similar phenomena too: in the Western world white = happiness (e.g., the color of a wedding gown), but in Japan white = death. In the Western world black = death, but in Japan black = happiness. A Black belt conjures in the western world the image of a potential killer, in Japan a black belt conjures up the image of someone who protects and brings happiness to the farmers).

I prefer to stay as close as possible to the tradition of double-entry bookkeeping, without giving up logical rigor, so I prefer to write A-L-E = 0. You want to redefine the valuations of the symbols L and E and write A+L+E=0. In both cases, we talk about a balance equation. That is not confusing. What is confusing is when the same words have two, exactly opposite, meanings, like "debit" and "credit".

The Accounting Equation and Risk

The following post was written and contributed by Cornelis ("Kees") A. Los, PhD, Professor of Finance. Faculty of Management, The University of Lethbridge (Alberta, Canada).

Reading your comments, it occurred to me that the "controversy" is only about how you define the terms A, L and E of the equation. Of course, it is useful to view liabilities (L) and equity (E) as negative assets (A), which is what the traditional Accounting Equation states A=L+E, thus A-L-E=0, a balance equation. What you did was defining first a balance equation A+L+E = 0, so that, in reverse, A=-(L+E) or L+E = -A. In the traditional case L and E are positive numbers, while they are negative numbers (negative assets) in your presentation.

In fact, that's how I programmed the assets and liabilities and equity of ING bank, when, as the Economic Advisor and Chief US Economist of ING bank, I proposed a financial risk management system, based on optimizing the ledger-portfolio of ING Bank in New York, using monthly departmental revenue and expense data. That project resulted in the elimination of ING bank's Chicago office, since it took too much risk and did not deliver enough return. But the Distressed Debt trading department was high-return-high-risk, while the Treasury was low-return-low-risk and that was fine, since both were lying on the Markowitz ledger-portfolio frontier.

That was a revelation to most ING bankers at that time (in 1993), since the Distressed Debt Department Head, who was also the Head Strategist of ING Bank in New York at that time, wanted to eliminate the Treasury because it did not earn enough on its invested capital. I argued that that would result in the elimination of the whole bank.....! My picture of ING Bank's ledger-portfolio frontier "saved" the Treasury of ING Bank in New York (true story!)

Computational Finance

The following post was written and contributed by Cornelis ("Kees") A. Los, PhD, Professor of Finance. Faculty of Management, The University of Lethbridge (Alberta, Canada).

An equation is what it is: it equates. in other words, what is at the left side of the equal (=) sign equals what is at the right side of that sign and it does not matter what is left or what is right, as long as they are equated. Therefore it is also called a balance equation.

The traditional Accounting equation can be written as A = L + E (= traditional accounting balance at one point in time, since A - L - E = 0). Modern finance, in particular financial engineering (e.g., options theory) has used this equation to compute a market value for equity E, when the assets (A) and liabilities are both valued at market prices: E = A - L.. Modern bankruptcy valuation, after the equity has "melted away" (E = 0) computes the value of the remaining liabilities by valuing the assets, so that L = A.

In a dynamic fashion, we can take differences of the entities involved: dA = dL + dE, which we can recognize as the net income (profit) equation: profits = dE = dA - dL = revenues - expenses.

With a simple manipulation, we can look at the (modified) duration equation: (dA/A) x A = (dL/L) x L + (dE/E) x E, which states that the relative sensitivity of the assets x total assets = relative sensitivity of the liabilities x liabilities + relative sensitivity of equity x equity. This useful equation becomes now a tool for financial risk analysis, since (dE/E) = (dA/A) x A - (dL/L) x L = (dA/A) x (E + L) - (dL/L) x L, so that the profit dE = (dA/A) + [(dA/A) - (dL/L)] x (L/E). Thus the profit depends on the relative sensitivities of the assets and liabilities and the leverage factor (L/E).

What sensitivities are we talking about? For example, banks are very sensitive to the level of the interest rate (r), so that we have (dA/dr) = (dL/dr) + dE/dr, i.e. the sensitivity of profits wrt. the interest rate dE/dr = (dA/dr)/A + [(dA/dr)/A - (dL/dr)/L] x (L/E). [Note: Bear Sterns' problem of a wipe-out was mostly its extremely high leverage factor of L/E close to 30 times, when its asset value A = its portfolio of loaned mortgages, was diminishing in value].

But another example would be to find the sensitivity of an institution wrt. the oil price (P): dE/dP = (dA/dP)/A + [(dA/dP)/A - (dL/dP)/L] x (L/E). When I was a Chief Economist of ING Bank for the US, in New York, in the early 1990s, I already computed the sensitivity of ING Bank's balance sheet wrt. the world oil price, as part of my duties as a financial risk analyst for ING Bank in New York. It is easy to imagine that many transportation companies are VERY sensitive to the oil price, so are agribusinesses and many manufacturing companies.

I have a discussion of this particular issue in my book Computational Finance, going all the way back to the Franciscan monk Luca Pacioli's original (1494) mathematical treatise on double-entry bookkeeping (called in Latin: Summa de Arithmetica, Geometria Proportioni et Proportionalita = Summary of Arithmeticvs, Geometry of Proportions and the Abacus, in English. Thus Pacioli clearly meant double-entry bookkeeping to be a mathematical construct, in particular an equation). So let nobody tell us what was meant by double-entry bookkeeping! The mathematician Pacioli earned his keep by tutoring mathematics to the sons and daughters of wealthy merchants or Florence, so he was very familiar with the double-entry bookkeeping used by those merchants of Florence and Venice and he generalized it into the now familiar balance equation A = L + E, even though he did it in Latin words and used no algebraic notation.

My book is Computational Finance, World Scientific Publishing Co, Singapore, 2001 (there is a small misprint in equation 1.5 on page 24, where a sign is reversed). I now even daresay that this balancing equation, which forms the bedrock of modern capitalism and finance, formed the inspiration for Isaac newton's balancing equation of forces: sum of all forces = sum of all (masses x accelerations) = 0), which is the bedrock for physics. Accounting preceded (Newtonian) Physics by ca 150 years. I will make the error correction in the second edition of my book, to be published at the end of this year.

6. General Ledger

To fully understand the meaning of the Accounting Equation, we need to reduce it to its most abstract level, determine its essence, and then reconstruct it to its most commonly recognizable form. As part of the “reduction to the abstract” process, let us begin by giving it an alternative form that will, perhaps, stimulate some critical thought. Let’s express the Accounting Equation as follows:

Toilet Paper Expense + Everything Else = 0

This form of the equation assumes that the organization keeps track of the amount of money that it spends on toilet paper by maintaining a separate expense account for that expenditure. The “Everything Else” term of the equation represents all of the accounts in the general ledger with the exception of the Toilet Paper Expense account. The Everything Else term includes accounts that are assets, liabilities, revenues, and other expense accounts. Together these two terms represent every account in the whole accounting system (the general ledger).

The meaning of this unorthodox expression of the Accounting Equation is simply this:

Since double-entry bookkeeping requires every deposit (debit) entry into one account be matched by a corresponding withdrawal (credit) entry from another account, the total balance of all the accounts will always be zero.

Essentially, this is all that the Accounting Equation says. The balance in the Accounting Equation is simply axiomatic of the process of double-entry bookkeeping. Every debit entry is matched by an opposing credit entry so that the balance of all the entries recorded remains zero.

If I know that the Toilet Paper Expense account has a debit (deposit) balance of $87.56, I also know that the balance of “Everything Else” must be $87.56 to the credit (withdrawal) side, allowing the total balance of the general ledger to be at zero.

Finally reduced to the most abstract level, the Accounting Equation says the following:

Regardless of how we partition the general ledger, the partitions must have a combined balance of zero – the balance of the general ledger itself.

Or, in mathematical terms, if we partition the accounts of the general ledger into “n” number of partitions (here, “n” represents any variable whole number), the following is true of the balances (B_i) of the partitions:

B₁ + B₂ + … + B_n = 0

This is all that the Accounting Equation is saying at the most essential level.

Now, to bring it back to the level of common practice, we can partition the accounts of the general ledger into terms that represent the company’s assets and other conceptual groupings:

Assets + Liabilities + Owners’ Equity + Temporary Accounts = 0

And, after we have closed the temporary accounts and assigned their balances to the Owners’ Equity term, this last equation becomes:

Assets + Liabilities + Owners’ Equity = 0

However, this very correct expression means something far different from the following:

Assets - ( Liabilities + Owners’ Equity) = 0

Which, although the traditional form of the Accounting Equation, is inconsistent with the most fundamental rules of double-entry bookkeeping.

5. Assets

As a thought experiment, let’s imagine an extremely simplistic accounting system. The system uses the double-entry bookkeeping method, but uses only two accounts, referred to as “Inside” and “Outside.” The Inside account represents all financial resources, regardless of use or liquidity, that are available for the company to use. The Outside account, of course, needs to represent all of the rest of the financial resources that are of interest to the company. More specifically, the Outside account represents resources, including property rights and obligations, which do not belong to the business (typically, they belong to owners, creditors and expenses recipients).

The company begins business as a blank sheet, with both accounts being empty and therefore having a balance of zero. Because both balances are zero, we can safely say that the combined balance of both accounts is also zero.

Inside + Outside = 0

As the company begins doing business, the balance of the Inside account changes. Regardless of whether the first transaction represents a contribution from the owner, a sale, or monies borrowed from a bank, it must have the effect of increasing the Inside account. Since the company does not have any resources to control, the first transaction cannot be a withdrawal from that account – it must be a deposit.

After the first transaction, the Inside account should have a balance that reflects a net depositing of resources (a debit balance, shown here as a positive amount).

Inside > 0

According to the rules of double-entry bookkeeping, however, the deposit in the Inside account must be matched by a corresponding withdrawal from the source account. In this simplistic example that can only come from the Outside account. At the end of the first transaction, the Outside account must have a balance that reflects a net withdrawing of resourced (a credit balance, shown here as a negative amount).

Outside < 0

And, since the withdrawal from the Outside account must be equal to the deposit in the Inside account, the balance of the general ledger itself (the whole system) must remain at zero.

Inside + Outside + deposit entry + withdrawal entry = 0

Or, simply:

Inside + Outside = 0

Each transaction after this first one will also keep the balance of the general ledger at zero. Further sales, loans, and contributions cause deposits to be recorded to the Inside and withdrawals to be equally recorded to the Outside. Expenses, loan repayments, and dividends cause withdrawals to the Inside account matched by deposits to the Outside account. Since the deposits are always equal in size to the withdrawals, the balance of the general ledger remains zero.

Inside + Outside + total deposits + total withdrawals = 0

Or, simply:

Inside + Outside = 0

Now, to bring our thought experiment closer to modern practice, we rename the Inside account “Assets” and our Outside account “Non-Assets,” we have the following:

Assets + Non-Assets = 0

And this mathematical expression of true balance is something very different from the traditional expression of the Accounting Equation, which, in our simplistic example would be expressed as:

Assets = Non-Assets

This final expression must always be false except in the degenerate case of when the Assets balance itself is zero.

4. Owners’ Equity

The Accounting Equation is essentially expressed as follows in all of the textbooks used to teach finance and accounting:

Assets = Liabilities + Owners’ Equity

This simple algebraic statement is actually stating that:

The more the owners invest in a business, the poorer they will be.

This obvious fallacy can be demonstrated by a simple example. For the sake of simplicity, let us take a simple startup business that has no liabilities. The owners have chosen to not borrow money and, as a startup, there are no accrued expenses that have not been paid. The sole financial transaction recorded by the company is the owners’ contribution of $100. Without any liabilities, we can reduce the accounting equation to the following:

Assets = Owners’ Equity

By the rules of accounting, the Owners’ Equity account (or one of it sub-accounts) must be credited with their contribution, indicating that the Owners’ Equity is the source of the $100 and that amount is effectively withdrawn from that account. That means that the balance of the Owners’ Equity is as follows:

Owners’ Equity = $100.00 (credit)

And, since our Accounting Equation states that:

Assets = Owners’ Equity

We can substitute the balance of the Owners’ Equity account ($100.00 credit) for the term itself, giving us:

Assets = $100.00 (credit)

But a credit balance for the Assets means that the company has had more resources withdrawn from it than it has had deposited into it. Because of the owners’ contribution of $100.00, the company’s asset value is negative.

The owners’ have made the company (and their stake in it) poorer by contributing to it.

3. Financial Balance

The previous post introduced the mechanics of double-entry bookkeeping and described how those mechanics kept the balance of the general ledger at zero.

General Ledger = 0

Again, double-entry bookkeeping is a matter of recording every transaction within an accounting system as both a deposit from someplace and a withdrawal from another. Since each withdrawal (“credit”) is the negative of a deposit (“debit”), the recording of both causes the balance of the whole system to remain at zero. The deposit increases the debit balance in one account but a correspond withdrawal from another account means that the system as a whole will always have a zero balance. While the individual accounts change their balances with each recorded transaction, the balance of the whole general ledger remains changed.

General Ledger + (Debit Entry + Credit Entry) = 0

Each recorded transaction has a zero effect on the general ledger itself. At the beginning of the business the general ledger started with a balance of zero and each transaction since that time has had a net effect of leaving that balance unchanged.

If we partition the accounts of the general ledger into various categories for whatever purpose that we may have, the sum of balances of all of the partitions remains zero. Like separate accounts, the partitions themselves may have different balances, but their total balance must be the same as the balance of the general ledger itself – zero. For example, if we partition all the accounts in the general ledger into permanent accounts and temporary accounts, we can assume the following:

Permanent + Temporary = 0

Since the Permanent and Temporary partitions together include all of the accounts in the general ledger (the definition of a partition), their combined balance must be the same as that of the general ledger itself.

Since the Accounting Equation is only valid at the time that the temporary accounts are closed and are all set to zero, the following is true when the Accounting Equation is valid:

Permanent + 0 = 0

Or:

Permanent = 0

The permanent accounts each belong to one of terms of the Accounting Equation, the assets, liabilities, or owner’s equity. These three terms are, effectively, a partition of the permanent accounts and, as such, have a combined balance that is the same as the balance of the permanent accounts:

Assets + Liabilities + Owner’s Equity = 0

And again, this zero equation is a true expression of balance, expressing that every deposit is balanced by a withdrawal and every transactions thereby cancels itself somewhere in the general ledger. This is also a very different expression from the general accepted accounting equation:

Assets = Liabilities + Owner’s Equity

In fact, this last expression of the Accounting Equation can only be true when a business has no asset value. For further details see The Tao of Financial Information:

2. What is Double-Entry Bookkeeping?

The meaning of the Accounting Equation must begin with a brief review of the foundation of all financial information – double-entry bookkeeping. Double-entry bookkeeping is the relating of each financial transaction to its source and destination (a financial transaction being a single event that results in the allocation of resources from one place to another).

The relating of the transaction to both its source and destination comprises the “double” in the term “double-entry.” Historically, each transaction has been related to each side of the exchange by having the bookkeeper make two recordings (“entries”) of the transaction, one in an account representing the source side of the transaction and a second one in an account representing the destination side.

The recording on the source side is known as a “credit” entry and is effectively a record of a withdrawal from an account representing that side. The recording on the destination side, in turn, is known as a “debit” entry and is effectively a record of a deposit into an account representing the destination side.

For example, when the business pays a utility bill, the payment transaction is recorded as a withdrawal (“credit”) from the cash account and a deposit (“debit”) to an account that represents utility expenses. The deposit increases the “debit” balance of the utility expense account while the withdrawal decreases the cash accounts “debit” balance. The amount reported as deposited in one account is exactly the same as the amount reported as withdrawn from another account.

Although a single transaction changes the balances in various individual accounts, it has a net effect of zero on the total balance of all of the accounts. Just as the utility expense account increased its debit balance in the previous example, the cash account’s balance was decreased by the same amount, causing absolutely no change to the combined balance of the two accounts together as well as the combined balance of the accounting system in general.

Each deposit is exactly balanced by its corresponding withdrawal, leaving the balance of the whole system (referred to as the “General Ledger”) at its point of origin – zero. The system begins with a zero balance and continues to have exactly a zero balance throughout its lifetime. Regardless of how many transactions we enter, each transaction’s recorded deposit is offset by its recorded withdrawal, leaving the balance of General Ledger just as it was before the first transaction was recorded – zero.

General Ledger = 0

And, if we partition the accounts of the General Ledger into two sets, set “A” and set “B”, we do not change its balance and therefore we have:

A + B = 0

In fact, regardless of how we partition the accounts of the General Ledger, their combined balance will be the balance of the General Ledger. If we partition the General Ledger into three sets representing the company’s assets, liabilities, and owner’s equity (after closing the temporary accounts), we have the following expression:

Assets + Liabilities + Owner’s Equity = 0

This formula is the correct way to express the Accounting Equation. It has a very different meaning from the standard way of expressing the equation:

Assets = Liabilities + Owner’s Equity

In following posts to this blog, the contradiction between these two expressions of the Accounting Equation and the significance of that contradiction to the financial world will be explored.

1. Financial Zero

This is the first post in a revolutionary blog that will reveal the true meaning of the Accounting Equation, the simple mathematical expression that serves as the cornerstone of our financial world. What is revolutionary about this blog is that it will show that, contrary to everything you may have been taught elsewhere, the only thing that the Accounting Equation expresses is that:

In double-entry bookkeeping, where each transaction is recorded as both a withdrawal from some account and a deposit to some other account, the balance of all the accounts must remain zero, the total withdrawals always canceling out the total deposits.

This, of course, is far different from the accounting equation found in most text books, where the accounting equation is expressed as follows:

Assets = Liabilities + Owner’s Equity

In following posts, this blog will demonstrate, incontrovertibly, the following:

1. The textbook version of the equation is mathematically invalid.

2. The equation only tells us that the balance of all accounts is zero.

3. Correcting the equation makes it more useful.

For further information, see chapter three of The Tao of Financial Information.