The document explains the concepts of stocks and flows in accounting, economics, and finance, emphasizing their importance for understanding financial statements. It details how stock measures represent values at a specific point in time, while flow measures represent values over a period, illustrating these concepts with equations and examples. Additionally, it describes the balance sheet and income statement as key financial statements, highlighting their roles in reporting a business's financial performance.

1. Basic math tools:
Stocks, flows, and financial statements
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August 30, 2011

2. Topics
Stocks and flows
Accounting fundamentals

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

4. Stocks and flows
Consider this figure:
It shows a container with a certain liquid (e.g. water), one pipe
delivering it into the container, and a pipe leaking it out.
Physicists and engineers call it a simple “hydrodynamic system.”

5. Stocks and flows
Basically, we can measure the performance of this system by
measuring
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
the amount of water that flows through the container over
given periods of time (e.g. a week, a month).
Then, by combining both forms of measurement, we get a richer
view of the functioning of the system. The stock measures may
help us correct errors in the measurement of flows or, vice versa.
(The term stock has several meanings in economics. It also refers
to the “equity” or legal ownership of a company traded in the
market. The context should make it clear when we use the term in
one or the other sense.)

6. The algebra of stocks and flows
Let the upper-case letters denote stocks and the lower-case ones
flows. Let Xt be the amount of water in the vessel at point in time
t, xit the amount of water flowing into the vessel during the
month, and xot the amount of water flowing out of the vessel
during the month. Then:
Xt+1 = Xt + xit − xot
The amount of water in the container at t + 1 is equal to the
amount of water in the container at t plus the water that got in it
during period (t, t + 1) minus the water that leaked out during the
same period.1
1
Assume no evaporation or, alternatively, that the water leaked out in the
period includes evaporated water.

7. The algebra of stocks and flows
Example: On November 1 (t = 0), the amount of water in the
container is 10 gallons (X0 = 10). The water flowing into the
vessel during November is 40 gallons (xit = 40) and the water
flowing out of the vessel is 39 gallons (xot = 39). Calculate the
amount 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 (greater
than or less than) the result above, we have made errors in our
measurements. The new measure of the water level can help us
correct them.

8. Some accounting
The two basic financial statements that accountants produce are
the balance sheet and the income statement. These financial
statements provide a detailed picture of the ongoing financial
performance of a business.

9. Balance sheet
The balance sheet of any organization (e.g. a business) has two
sides. The left-hand side reports the value of all the organization’s
assets at a given point in time, typically at the end of a year. The
assets are the resources the organization manages measured at a
point in time. The right-hand side reports the source of the assets
listed 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), A
is total assets, L is total liabilities, and E is total equity.2
2
To separate an organization from its individual owners, it may be
convenient to state the equation as A = L, i.e. assets equal liabilities. They are
liabilities to either others or to the individual “owners” of the organization. In
this interpretation, the equity of the legal owners of, e.g., a business is
considered a special type of liability.

10. A typical balance sheet
ABC Inc.
Balance Sheet
12/31/06
Assets Liabilities and Equity
Cash and liquid securities $ 10 Payables $ 20
Inventories 50 Other short-term debt 40
Receivables 60 Mortgages 80
Trucks (net) 25 Other long-term debt 250
Office equipment (net) 10 Liabilities 390
Machinery (net) 45 Equity 120
Buildings (net) 220
Other fixed assets (net) 90
Assets $ 510 Liabilities plus Equity $ 510

11. Income statement
The income statement reports on its top line the total flow of gross
income (e.g. sales revenues) received by the organization during a
period of time (e.g. a year). The next lines report the various
expenses that the activity of the organization incurred during the
period to sustain its gross income. These expenses – sorted out as
production costs, operating expenses, financial expenses, and taxes
– are deducted or subtracted from the top line. The bottom line of
the income statement indicates the flow of net income or net profit
during 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 in
time t, NI is the residual or net income (profit if positive, loss if
negative), GI is the gross income (typically, sale revenues), PC is
the total cost of goods sold (e.g. the cost of raw materials, storage
costs, wages and benefits of factory-floor workers), OE are the
operating expenses (sales and administrative expenses, including
salaries and commissions of administrative and sales personnel,

12. Income statement
A typical income statement:
ABC Inc.
Income Statement for
1/1/07 through 12/31/07
Sales revenues $ 200
(-) Cost of goods sold 90
Gross profit 110
(-) Operating expenses (includes depreciation) 40
Operating profit 70
(-) Interest paid 10
Taxable profit 60
(-) Taxes 10
Net profit $ 50

13. Summary
Note that:
all items in the balance sheet are stock measures (“water in a
container” at a point in time),
all items in the income statement are flow measures (“water
that flows in/our” over a period of time),
the balance sheet and the income statement are related in
multiple ways,
assets are “positive water stocks” while liabilities and equity
items are “negative water stocks” (if that makes sense), and
sale revenues are “positive water flows” and costs and
expenses are “negative water flows’.’

14. Summary
Every time an organization conducts an operation or transaction,
every time a business takes raw materials from its inventories and
have its workers process them on the factory floor, every time its
sales people sell a batch of goods or its administrative personnel
orders a shipment from its suppliers, every time a payment is made
or received, there is “water” flowing from one balance-sheet
container into another one. At the end of the given period (and
beginning of the next period), the balance sheet reports the
adjusted levels of “water” in each container at that point in time.
Also at the end of the given period (beginning of the next), each
spurt of “water” that flowed from container to container during
the period is added up (aggregated) into its respective category
and recorded in the income statement. The legal owners of the
organization (if a corporation, the legal owners are called
stockholders) pay most attention to the level of “water” in their
equity “container.”