Basic math tools:Stocks, flows, and financial statements

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August 30, 2011

Topics

Stocks and flows

Accounting fundamentals

Stocks and flows

In economics (as in the physical sciences), empirical observationsand measurements are made either at a point in time or over aperiod of time. As a result, the basic variables can be eitherstocks or flows. A stock measure refers to the value of a variableat a given point in time. A flow measure refers to the value of avariable over a given period of time. On top of these basicmeasures, a host of derivative measures (e.g. ratios of flows toflows, flows to stocks, stocks to flows, and stocks to stocks) canbe determined. This distinction is extremely useful in accounting,economics, and finance.

Stocks and flows

Consider this figure:

It shows a container with a certain liquid (e.g. water), one pipedelivering it into the container, and a pipe leaking it out.Physicists and engineers call it a simple “hydrodynamic system.”

Stocks and flows

Basically, we can measure the performance of this system bymeasuring

I the amount of liquid in the container at given points in time(e.g. on Sundays at noon, on the last day of each month) and

I the amount of water that flows through the container overgiven periods of time (e.g. a week, a month).

Then, by combining both forms of measurement, we get a richerview of the functioning of the system. The stock measures mayhelp us correct errors in the measurement of flows or, vice versa.(The term stock has several meanings in economics. It also refersto the “equity” or legal ownership of a company traded in themarket. The context should make it clear when we use the term inone or the other sense.)

The algebra of stocks and flows

Let the upper-case letters denote stocks and the lower-case onesflows. Let Xt be the amount of water in the vessel at point in timet, xit the amount of water flowing into the vessel during themonth, and xot the amount of water flowing out of the vesselduring the month. Then:

Xt+1 = Xt + xit − xot

The amount of water in the container at t + 1 is equal to theamount of water in the container at t plus the water that got in itduring period (t, t + 1) minus the water that leaked out during thesame period.1

1Assume no evaporation or, alternatively, that the water leaked out in theperiod includes evaporated water.

The algebra of stocks and flows

Example: On November 1 (t = 0), the amount of water in thecontainer is 10 gallons (X0 = 10). The water flowing into thevessel during November is 40 gallons (xit = 40) and the waterflowing out of the vessel is 39 gallons (xot = 39). Calculate theamount of water in the container on December 1:

X1 = X0 + xi1 − xo1 = 10 + 40 − 39 = 11.

If the stock of water on December 1 is any different from (greaterthan or less than) the result above, we have made errors in ourmeasurements. The new measure of the water level can help uscorrect them.

Some accounting

The two basic financial statements that accountants produce arethe balance sheet and the income statement. These financialstatements provide a detailed picture of the ongoing financialperformance of a business.

Balance sheet

The balance sheet of any organization (e.g. a business) has twosides. The left-hand side reports the value of all the organization’sassets at a given point in time, typically at the end of a year. Theassets are the resources the organization manages measured at apoint in time. The right-hand side reports the source of the assetslisted on the left-hand side. They are either owed to others(liabilities) or “owned” by the legal “owners” (equity or net worth).The fundamental balance-sheet equation is:

At = Lt + Et

where t indicates a point in time (e.g. the last day of the year), Ais total assets, L is total liabilities, and E is total equity.2

2To separate an organization from its individual owners, it may beconvenient to state the equation as A = L, i.e. assets equal liabilities. They areliabilities to either others or to the individual “owners” of the organization. Inthis interpretation, the equity of the legal owners of, e.g., a business isconsidered a special type of liability.

A typical balance sheet

ABC Inc.Balance Sheet12/31/06

Assets Liabilities and Equity

Cash and liquid securities $ 10 Payables $ 20Inventories 50 Other short-term debt 40Receivables 60 Mortgages 80Trucks (net) 25 Other long-term debt 250Office equipment (net) 10 Liabilities 390Machinery (net) 45 Equity 120Buildings (net) 220Other fixed assets (net) 90

Assets $ 510 Liabilities plus Equity $ 510

Income statementThe income statement reports on its top line the total flow of grossincome (e.g. sales revenues) received by the organization during aperiod of time (e.g. a year). The next lines report the variousexpenses that the activity of the organization incurred during theperiod to sustain its gross income. These expenses – sorted out asproduction costs, operating expenses, financial expenses, and taxes– are deducted or subtracted from the top line. The bottom line ofthe income statement indicates the flow of net income or net profitduring the period. The fundamental equation is:

NIt = GIt − PCt − OEt − FEt − Tt

where t is the period of time from point in time t − 1 to point intime t, NI is the residual or net income (profit if positive, loss ifnegative), GI is the gross income (typically, sale revenues), PC isthe total cost of goods sold (e.g. the cost of raw materials, storagecosts, wages and benefits of factory-floor workers), OE are theoperating expenses (sales and administrative expenses, includingsalaries and commissions of administrative and sales personnel,depreciation of buildings and equipment), FE are the financialexpenses (interest payments), and T are the taxes.

Income statement

A typical income statement:

ABC Inc.Income Statement for1/1/07 through 12/31/07

Sales revenues $ 200

(-) Cost of goods sold 90Gross profit 110(-) Operating expenses (includes depreciation) 40Operating profit 70(-) Interest paid 10Taxable profit 60(-) Taxes 10

Net profit $ 50

Summary

Note that:

I all items in the balance sheet are stock measures (“water in acontainer” at a point in time),

I all items in the income statement are flow measures (“waterthat flows in/our” over a period of time),

I the balance sheet and the income statement are related inmultiple ways,

I assets are “positive water stocks” while liabilities and equityitems are “negative water stocks” (if that makes sense), and

I sale revenues are “positive water flows” and costs andexpenses are “negative water flows’.’

Summary

Every time an organization conducts an operation or transaction,every time a business takes raw materials from its inventories andhave its workers process them on the factory floor, every time itssales people sell a batch of goods or its administrative personnelorders a shipment from its suppliers, every time a payment is madeor received, there is “water” flowing from one balance-sheetcontainer into another one. At the end of the given period (andbeginning of the next period), the balance sheet reports theadjusted levels of “water” in each container at that point in time.Also at the end of the given period (beginning of the next), eachspurt of “water” that flowed from container to container duringthe period is added up (aggregated) into its respective categoryand recorded in the income statement. The legal owners of theorganization (if a corporation, the legal owners are calledstockholders) pay most attention to the level of “water” in theirequity “container.”