Deriving Average Balance Denominators

Why Average a Balance?

Many financial ratios compare a flow measured over a period with a balance measured at a date. For example, cost of goods sold is accumulated over a year, while inventory is reported at a particular date. The beginning-and-ending average gives an estimate of the resources held, on average, while the period’s activity occurred.

This is an estimate, not a claim that the balance stayed constant or changed smoothly. Its usefulness depends on how representative the two reported endpoints are of the balances during the period. To use the method well, distinguish the balance at the start, the balance at the end, and the flow for the period.

Definition: For a balance-sheet amount with beginning balance $B_0$ and ending balance $B_1$, the beginning-and-ending average balance is $$ \bar{B}=\frac{B_0+B_1}{2}. $$ It estimates the average amount of that resource or claim during the period. A ratio that uses it as a denominator pairs a period flow with this estimated average stock.

Deriving the Average From a Balance Path

The arithmetic average has a precise interpretation under a simple model: suppose the balance changes at a constant rate from the beginning to the end of the period. Let $t$ measure elapsed time, with $0\leq t\leq T$, and let the balance at time $t$ be $B(t)$. Under this linear-change assumption,

B(t)=B_0+\frac{B_1-B_0}{T}t.

The time-average balance is the total balance-time exposure divided by the length of the period. Integrating the assumed path gives:

\begin{aligned} \bar{B} &=\frac{1}{T}\int_0^T B(t)\,dt\\ &=\frac{1}{T}\int_0^T\left(B_0+\frac{B_1-B_0}{T}t\right)dt\\ &=\frac{1}{T}\left(B_0T+\frac{B_1-B_0}{T}\cdot\frac{T^2}{2}\right)\\ &=B_0+\frac{B_1-B_0}{2}\\ &=\frac{B_0+B_1}{2}. \end{aligned}

The result is the midpoint of the two endpoints because a straight-line path spends equal time at equal increments along that path. The derivation also identifies the assumption: if the balance moves sharply during the period, the endpoints alone may not describe the average well. A business with seasonal inventory, for instance, could have an average quite different from the midpoint of its year-start and year-end amounts.

Key result: The beginning-and-ending average is exact for a balance that changes linearly over time. When the actual path is unknown, it is a practical estimate based on the available opening and closing balances; it is not automatically the exact average.

Using an Average Balance in a Ratio

The basic calculation has two parts: average the relevant balance, then divide the flow for the period by that average. The particular numerator and denominator depend on the question being answered. Here the focus is on calculating the average denominator correctly, not on treating every ratio as interchangeable.

\text{Ratio}= \frac{\text{flow for the period}}{\text{average balance during the period}}, \qquad \text{where}\qquad \text{average balance}= \frac{\text{beginning balance}+\text{ending balance}}{2}.

For an inventory turnover calculation, for example, the flow is cost of goods sold and the balance is inventory. The ratio expresses the period’s cost of goods sold relative to the inventory held on average. For return on assets, the period’s net income is compared with average total assets. In each case, the average balance gives the denominator a period-wide interpretation rather than describing only one date.

Worked Example: Inventory Turnover With Unequal Opening and Closing Balances

A hypothetical distributor reports beginning inventory of $180,000 and ending inventory of $220,000 for its fiscal year. Cost of goods sold for the year is $1,200,000. Calculate inventory turnover using the beginning-and-ending average. State what the result represents and check the arithmetic.

Method. First calculate the estimated average inventory:

\begin{aligned} \text{Average inventory} &=\frac{\$180{,}000+\$220{,}000}{2}\\ &=\frac{\$400{,}000}{2}\\ &=\$200{,}000. \end{aligned}

Next divide the annual cost of goods sold by average inventory:

\text{Inventory turnover} =\frac{\$1{,}200{,}000}{\$200{,}000} =6.0\text{ times}.

The result says that annual cost of goods sold is six times the estimated average inventory balance. As a check, multiplying the denominator by the reported turnover recovers the numerator: $6.0\times \$200{,}000=\$1{,}200{,}000$. The common mistake this method avoids is dividing by the ending inventory alone: that would give $\$1{,}200{,}000/\$220{,}000\approx 5.45$, a different calculation that does not use the estimated period-average balance.

Worked Example: Return on Assets Using Average Assets

A hypothetical service firm has total assets of $1,100,000 at the start of the year and $1,300,000 at year-end. Its net income for the year is $96,000. Calculate return on assets using the beginning-and-ending average, then compare it with the result from using only ending assets.

Method. The average total assets are:

\begin{aligned} \text{Average total assets} &=\frac{\$1{,}100{,}000+\$1{,}300{,}000}{2}\\ &=\frac{\$2{,}400{,}000}{2}\\ &=\$1{,}200{,}000. \end{aligned}

Using net income as the period numerator:

\text{Return on assets} =\frac{\$96{,}000}{\$1{,}200{,}000} =0.08=8.00\%.

For comparison, using ending assets alone would produce:

\frac{\$96{,}000}{\$1{,}300{,}000} =0.073846\ldots \approx 7.38\%.

The average-balance calculation is 8.00%; the ending-balance calculation is approximately 7.38%. They differ because the denominator differs, not because the net income changed. As a check, $8.00\%\times \$1{,}200{,}000=\$96{,}000$. The common mistake is to assume that a date-specific ending figure must stand for the entire year, even when the chosen method calls for an average.

Worked Example: Receivables Turnover With an Average Denominator

A hypothetical equipment lessor begins the year with accounts receivable of $140,000 and ends with accounts receivable of $220,000. Credit revenue for the year is $2,400,000. Calculate receivables turnover using average accounts receivable and verify the result.

Method. Calculate the average receivables balance:

\begin{aligned} \text{Average accounts receivable} &=\frac{\$140{,}000+\$220{,}000}{2}\\ &=\frac{\$360{,}000}{2}\\ &=\$180{,}000. \end{aligned}

Divide credit revenue for the year by that average:

\text{Receivables turnover} =\frac{\$2{,}400{,}000}{\$180{,}000} =13.3333\ldots \approx 13.33\text{ times}.

Thus credit revenue is about 13.33 times the estimated average receivables balance. To check, $13.3333\ldots\times \$180{,}000=\$2{,}400{,}000$. The common mistake is to average the flow and the stock together or to use their difference; neither produces the required denominator. The denominator is the arithmetic average of the two receivables balances, and the numerator remains the full-period credit revenue.

A Reliable Calculation Process

A consistent sequence helps prevent errors when opening and closing balances are presented among several statement amounts. Keep the balance calculation separate from the ratio calculation, and label each value before dividing.

Identify the relevant balance.
Use the balance-sheet account that represents the resources or claims named by the ratio, and record its beginning and ending amounts.

Calculate the average.
Add the beginning and ending balances and divide their sum by two. Keep the units, such as dollars or thousands of dollars, visible.

Pair it with the period flow.
Divide the relevant flow for the period by the average balance, preserving the ratio’s intended meaning.

Check the result.
Multiply the calculated ratio by its denominator and confirm that the result recovers the numerator, subject to any displayed rounding.

Assumptions, Limits, and a Common Trap

The arithmetic average is simple, but its interpretation should remain precise. Under a linear balance path, it equals the time-average balance. In practice, a ratio calculation may use the same midpoint when only beginning and ending figures are provided, while recognizing that the midpoint is an estimate of the resources used over the period.

If the balance changes irregularly, a two-point average can miss important movements. For example, a balance could rise substantially during the year and then fall back near its opening amount; the beginning and ending figures would not reveal how high it was in between. If suitable interim balances are available, an analyst can instead calculate an average using those observations. The chosen method should be applied consistently and interpreted in light of the information available.

Do not substitute the ending balance for an average without noticing. A ratio using an ending balance is mathematically different from one using a beginning-and-ending average.
Do not average unrelated accounts. The denominator must be the balance specified by the ratio being calculated.
Do not divide by two after dividing. The two balances are added first; their sum is divided by two to form the denominator.
Do not describe the estimate as a directly observed daily average. The midpoint uses two dates and, by itself, does not reveal the path between them.

The central reasoning is that a period flow reflects activity accumulated across time, whereas a balance-sheet amount is observed at a point in time. Averaging the opening and closing balances supplies a simple estimate of the stock supporting that activity over the period. When this estimate is used, show the two balances and the average explicitly: doing so makes the denominator transparent and makes errors easier to detect.

Check Your Understanding

For each problem, calculate the average balance before calculating the requested ratio.

A hypothetical food processor reports beginning inventory of $310,000, ending inventory of $290,000, and annual cost of goods sold of $1,800,000. Calculate inventory turnover using average inventory. (Answer: average inventory is $300,000; turnover is 6.00 times.)
A hypothetical software company has beginning total assets of $2,400,000, ending total assets of $3,000,000, and annual net income of $216,000. Calculate return on assets using average total assets. (Answer: average assets are $2,700,000; return on assets is 8.00%.)
A hypothetical repair network has accounts receivable of $260,000 at the beginning of the year and $340,000 at year-end. Annual credit revenue is $4,200,000. Calculate receivables turnover using average receivables. (Answer: average receivables are $300,000; turnover is 14.00 times.)
A hypothetical wholesaler has beginning inventory of $450,000 and ending inventory of $550,000. Annual cost of goods sold is $3,000,000. Calculate turnover using the beginning-and-ending average, then calculate the result using ending inventory alone. (Answer: average inventory is $500,000; average-balance turnover is 6.00 times; ending-balance turnover is approximately 5.45 times.)
A hypothetical firm reports beginning assets of $800,000, ending assets of $1,000,000, and annual net income of $72,000. An analyst uses only ending assets. Calculate return on assets using the average-balance method and quantify the percentage-point difference from the ending-balance result. (Answer: average assets are $900,000; average-balance return on assets is 8.00%; ending-balance return is 7.20%; the average-balance result is 0.80 percentage points higher.)

What You'll Learn