Relational Matrix as the Ultimate Financial Statement

The accounting profession has lost sight of the true nature of double-entry bookkeeping and this gravely limits the analytical power of today's financial record keeping. Double-entry bookkeeping is far more than providing a duplicated check on our recording accuracy. Each of the two entries provides unique and non-duplicated information about the transaction. The credit entry tells us the origin of the resources being transferred and the debit entry tells us the destination of those resources.

Once we realize the importance of each entry we can begin to harness the tremendous analytical power that it provides. The real financial statement for providing business behavior is not the income statement or the balance sheet; the real tool for analysis is the relational matrix.

The relational matrix has a column for all the accounts and a row for all of those accounts. The columns can hold the record of the debits and the rows can hold the record of the credits. Each cell of the table represents a transfer from the row that is credited to the column that is debited. A given cell tells us the sum that has gone from the corresponding row's account to the receiving column's account.

The sums of the rows and columns contains the information in the balance sheet. The individual cells represent the flow of resources from the one account to another (for graphical examples see my book, The Tao of Financial Information).

Accounting as Data Warehousing

The accounting journal is nothing more than a simple data warehouse. The direction of the transfer or resources in a transaction (from a credited account to a debited account) is only one dimension of a potentially multi-dimensional database, allowing the analyst to to view financial behavior through many different critical factors.

Another connection that needs to be made between the accounting world and the data processing industry is the recognition that the accounting journal is a powerful and complete form of event processing. Each transaction recorded in the journal is an event that is part of the history of a business organization.

The ability to do financial analysis is stunted by the inability of the accounting and IT professions to cross-fertilize their respective skills.

Vector Power

To fully appreciate the elegance of double-entry accounting, one needs to realize that each set of double entries defines an economic vector, tracking the movement of financial resources from one place to another. The credit entry identifies the origin of the transfer and the corresponding debit entry identifies the destination of the resources.

The insight that the credit/debit entry is a vector can greatly empower (and simplify) financial analysis and can allow us to enhance the record-keeping potential of the accounting journal. Accounting is the tracking of resource allocation through time. The economic vectors recorded by double-entry accounting can track the allocation of resources through any number of dimensions (for example; geography, business units, corporate organization, and consumer demographics).

For further explanation of the vector approach, obtain a free copy of the book, Banking on the Past, by adding a comment here or emailing this author at rob.meldahl@gmail.com.

Vector Accounting

I want to thank Irus, who added a comment to an earlier post, for the use of the term "vector" to describe the approach that I am using to clarify the process of accounting. A vector is a quantity that has both a magnitude and a direction. This is exactly what a debit or credit entry is. The magnitude is the monetary amount of the entry and the direction is either "in" (debit) or "out" (credit). Accountants have been recording vectors into data warehouses for five centuries; they just did not know it and have failed to use the resulting powers.

For more on vector accounting, see the blog at http://taofinancial.blogspot.com/. Also, get a free electronic copy of my two books, The Tao of Financial Information and Banking the Past, by sending me an email at rob.meldah@gmail.com.

8. Debit and Credit

To fully understand the accounting equation and financial data in general, we need to have an accurate definition of the terms “debit” and “credit.” These two terms are historical and their origins are of semantic significance. What a bookkeeper does when he records a transaction with debit and credit entries goes to the very essence of financial information.

For starters, debit and credit are not Latin words for “left” and “right,” despite claims of various textbooks. The whole financial world would work unchanged if debits were recorded on the right rather than the left. Within the computers that process accounting software, left and right have no real meaning. If debit and credit have any significance in producing critical and accurate financial information, they are not substitute terms for left and right.

And, despite authoritative claims to the contrary, debit and credit do not change their meaning as we go from the left side of the accounting equation to its right.

Assets = Liability + Equity

A debit to the Assets side of this equation behaves exactly like a debit to the Liability term on the far side of the equal sign. It increases a debit balance equally on both sides; it decreases a credit balance exactly the same in both places; and in all cases, a debit represents an increase in the amount of financial resources available at the location to which it is applied. Most importantly, the terms debit and credit were used exact as they are today for several hundred years before the Accounting Equation did. The two terms are in no way defined by the Accounting Equation itself.

Returning to the Latin sources of the terms, they have simple meanings that describe exactly their role in double-entry bookkeeping and reveal exactly the underlying meaning of all financial data. A debit is a Latin term that serves as a root for our modern term “debt.” It represents a deposit of something from some other place. Its original use in accounting and its use in modern financial data remain consistent with its original Latin meaning; a debit is a deposit of financial resources, regardless of which side of the accounting equation and which side of the t-account that it is recorded upon.

In turn, credit is derived from a Latin term meaning the source of a deposit. This also is consistent with modern bookkeeping entries that credit revenue, equity, and liability accounts when they serve as sources to deposits to cash and other asset accounts. This is also consistent with the application to credit entries to the cash account when that account serves as the source of deposits to expense accounts

Accounting debits and credits are just records of deposits and withdrawals as financial resources flow from one place to another in a defined financial space. As a withdrawal, a credit is the opposite of a debit and numerically is accurately used as a negative debit in parts of the accounting process.

The problem is that the polar opposition between debit and credit is often obscured by the inaccurate application of the Accounting Equation, rendering modern accounting fragmented and subject to unnecessary rules that need to be learned by rote memory. Correcting the arithmetic in the Accounting Equation would make all financial data simpler and more accessible to the average person.

Assets + Liabilities + Equity = 0

And this, in turn, would make critical financial information more available to those who allocate the financial resources of our economy.

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".