Mathematics and accounting

Summary: Accounting applies mathematics to the recording and analysis of a business’s financial status.

Accounting is the recording, interpretation, and presentation of financial information about a business entity, typically with the goal of producing financial statements that describe the business’s economic resources in standardized terms. Formal accounting began with the work of Franciscan friar Luca Pacioli, who introduced accounting techniques in his 1494 mathematical work Summa de Arithmetica, Geometria, Proportioni et Proportionalita. During the Industrial Revolution, Josiah Wedgwood introduced cost accounting, a technique to ensure a profit margin by calculating the costs of materials and labor at every stage of production and setting the price accordingly. The needs of stockholders and other interested parties within the business, and an increasingly complex business environment, have increased the need for financial record-keeping techniques that are thorough and produce useful financial statements. Modern accounting is assisted by a variety of software packages, but the accountant must still be well-versed in mathematics in order to interpret the information. The fundamental accounting equation can be stated as the following:

Assets = Liabilities + Owners’ Equity.

For any given company, assets can be thought of as what the company owns. This includes cash (actual cash and bank accounts), money that is owed to the business (called accounts receivables), inventory, buildings, land, equipment, and intangibles like patents and goodwill. Liabilities are what the company owes. This includes money owed to a bank (notes payable), suppliers (accounts payable), or the government (taxes payable). Owners’ equity can take several forms depending on who the owners are: a single person (sole proprietor), a few people (partnership), or shareholders (corporation). Each method of ownership has advantages and disadvantages, but regardless of the method, the owners’ equity can be thought of as a net asset since it can be found by subtracting liabilities from assets.

Accounting as Record Keeping

Whenever a financial transaction takes place, it must be recorded in at least three locations. First, it will be recorded in the general ledger (a book of entry summarizing a company’s financial transactions). When recorded, the entry should contain the date of the transaction, a brief description of the transaction, and the monetary changes to all accounts affected (which will be at least two).

From there, the transaction gets recorded a second time in a secondary (or subsidiary) ledger for each of the accounts affected. When the amounts are recorded, they are put into the left (debit) column or the right (credit) column of the ledger. (In bookkeeping, “debit” and “credit” mean left and right, respectively; they are not related to debit or credit cards in this situation.) The total of each column of the general ledger record must add to the same sum. In that manner, all money can be accounted for as going into or out of an account.

In order to determine whether to credit or debit an account, a general rule that works for most accounts is to first look at the fundamental accounting equation. Since assets are listed on the left, to increase assets, the transaction is recorded in the left column (debiting the account) and to decrease assets, the transaction is recorded in the right column (crediting the account). Similarly, since liabilities and owners’ equity are listed on the right-hand side of the equation, to increase liabilities and owners’ equity, the transaction is recorded in the right column (crediting the account) and to decrease liabilities and owners’ equity, the transaction is recorded in the left column (debiting the account). For example, suppose a company needed to purchase $100 worth of office supplies. Furthermore, suppose the company pays $40 with cash and puts the remaining $60 on account (store credit). The general ledger may look like the following:

Figure 1. Purchased office supplies.

In Figure 1, notice that both the right and left columns add up to $100; this shows that no money was lost in the process. Office supplies are considered an asset, so since the company increased the amount of office supplies, that account was recorded on the left—in other words, it debited office supplies for $100. Cash is also an asset, but the company decreased the amount of cash it had. As a result, cash was credited (the transaction was recorded on the right for that account). Accounts payable is a liability the company owes to the retailer it purchased the products from. Since the company increased the amount it owed the retailer, that account was recorded on the right as an increase to the company’s liabilities—accounts payable was credited.

Once this transaction was recorded in the general ledger, the company would also need to record this transaction in the Office Supplies ledger, the Cash ledger, and the Accounts Payable ledger. Accounts are debited or credited in their specific ledgers in the exact same manner that they are debited or credited in the general ledger. In a similar manner, the retailer who sold the office supplies would need to record this same transaction into his or her general and secondary ledgers. However, the retailer’s transaction would use the opposite side to denote the sale as follows:

Figure 2. Sold office supplies.

Again, the right and left columns add up to the same amount. Contrary to the purchasing company, the receiving company lists three assets to record the transaction. Cash and accounts receivable are both being increased, so debited. The asset “inventory” is being decreased and results in a credit to inventory. If this were a large company, rather than record each individual transaction, the retailer would most likely record an entire day’s transactions as a single entry at the end of each business day. Once the general ledger has been recorded, the secondary ledgers need adjusting entries as well to denote the transaction(s).

Accounting as Record Sharing

In addition to keeping records of transactions for a business, accounting is responsible for creating reports that summarize the journals to share with others. To learn about the reports and how to create reports intended for people outside the business (such as shareholders, creditors, or government agencies), a person can take a class in financial accounting. To learn about the reports and how to create reports intended for people inside the business (such as managers), a person can take a class in managerial accounting.

The most common reports created for people outside the business are balance sheets, income statements, cash flow statements, and retained earnings statements. Of the four statement types, the balance sheet is written as a snapshot of the company at a point in time. In contrast, the other three statements are created to show what happened over a period of time such as a month, quarter, or year. When creating these reports, the income statement is usually completed first. As its name implies, the income statement is created to determine the company’s income during a specific time period. The income statement is also known as a profit and loss statement (P&L) or earnings statement. Information from the income statement is then used to create the retained earnings statement. Finally, the information from the retained earnings statement is used on the balance sheet.

The balance sheet first lists all of the company’s assets in order of liquidity (the ability to turn the asset into cash easily) from the most liquid to the least liquid. The assets are then added together to find the total assets of the company. The balance sheet next lists all of the company’s liabilities in order of due date from the soonest due to the latest due. Below the liabilities is listed the owners’ equity (which includes retained earnings from the retained earnings statement). The liabilities and owners’ equity are added together. Referring back to the fundamental accounting equation, both of these amounts (the total assets and the sum of the liabilities and owners’ equity) should equal one another.

Reports created for internal users vary widely depending on the reasoning and the need for the report. Internal reports are usually created and specifically designed for making decisions within the company. For example, manufacturers could use internal reports to determine the optimal price of their product.

Manufacturers may also use internal reports to determine if it is more cost effective to create a needed part or to purchase the part from another company. They may need to consider continuing or eliminating a division of their company. Managerial accounting is also responsible for budgeting and forecasting.

Mathematical Models

Many areas in financial accounting rely on mathematical models for explanation and prediction. For example, models have played important roles in applications such as understanding the consequences of public disclosure, formalizing market efficiency or competition, measuring income, and evaluating equilibrium pricing for goods and services. Some important mathematical techniques used in accounting models include linear regression, systems of simultaneous equations, equilibrium notions, and stochastic analysis. In the latter, random rather than constant inputs are used to model scenarios where decisions must be made under realistic conditions of uncertainty. The data used in these models may be cross-sectional (representing a single snapshot in time) or longitudinal (one or more variables are measured repeatedly to detect trends and patterns). Probability theory is also used to detect instances of accounting fraud.

Bibliography

Davis, Morton D. The Math of Money: Making Mathematical Sense of Your Personal Finances. New York: Copernicus, 2001.

Hoyle, Joe Ben, Thomas F. Schaefer, and Timothy S. Doupnik. Fundamentals of Advanced Accounting. New York: McGraw-Hill, 2010.

Kimmel, Paul D., Jerry J. Weygandt, and Donald E. Keiso. Financial Accounting: Tools for Business Decision Making. 5th ed. Hoboken, NJ: Wiley, 2009.

Mullis, Darrell, and Judith Handler Orloff. The Accounting Game: Basic Accounting Fresh From the Lemonade Stand. Naperville, IL: Sourcebooks, 2008.

Verrecchia, Robert. “The Use of Mathematical Models in Financial Accounting.” Journal of Accounting Research 20 (1982).

Weygandt, Jerry J., Paul D. Kimmel, and Donald E. Keiso. Managerial Accounting: Tools for Business Decision Making. 4th ed. Hoboken, NJ: Wiley, 2008.