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Introduction

This article arose from an actual litigation. To protect the identity of the parties, I use fictional names. Let's assume that a fire started in the manufacturing plant of Sir Harvard Cucumber's firm, Cucumber Pickles (CP) and spread to his next-door-neighbor, Billabong's Boomerangs (BB), owned by Constance Billabong.

BB sued CP for a variety of different damages. We will focus on the two main conceptual categories of damages: the cost of the destroyed boomerangs and the lost profits from lost sales of boomerangs. In turn, there were two categories of lost profits: those from the inventory that was produced and destroyed, and sales that were lost from inventory that was never produced, because the Company was unable to take orders for one day and did not manufacture the inventory for the orders that were not taken.

We will develop formulas for calculating both categories of damages. Additionally, we develop separate formulas for the two types of lost profits. It is convenient to begin this article with formulas for losses for a manufacturing firm, as it an easy simplification to modify the formulas for retail or wholesale firms.

Rather than developing the theoretical mathematics right away, it makes for a more intuitive understanding if we first look at some tables with damage calculations. We will go through the logic and the numbers in the tables, and then we will develop general formulas for the destroyed inventory and lost profits.

Commentary to Table 1: Sample Loss Calculations

The purpose of Table 1 is to present a simple set of facts for lost inventory and lost profits calculations and develop formulas that quantify the damages accurately.

The organization of the table is as follows. Columns B through D are the calculations of total damages. Column B shows a simple income statement without the fire, column C shows the same with the fire, and column D is the difference, which is the total damage. Columns F and G are a decomposition of the total damages into lost inventory (boomerangs) and lost profits.

Column B: Net Income Without the Fire

Let's begin with column B, which is a simple, hypothetical income statement for BB if there were no fire. Sales (denoted as "S" in cell A6) are $100 (cell B6). Variable manufacturing costs ("VM"), which are composed of direct materials, direct labor, and variable manufacturing overhead, are $95 (B8).2 Fixed manufacturing overhead ("FMOH") is $10 (B9). Thus, total manufacturing overhead is $105 (B10), with a gross profit of -$5 (B11). Selling expenses are $10 (B12), and general, and administrative (G&A) expenses are $5 (B13), for a total SG&A expense of $15 (B14). BB's losses equal the negative $5 gross profit minus SG&A expense of $15, for a total of -$20 (B15).

Column C: Net Income With the Fire

Column C is net income with the fire. Sales are zero (C6) instead of $100. Variable and manufacturing costs and fixed manufacturing overhead are still $95 and $10 (C8 and C9). Thus, gross profit is -$105 (C11). Selling expenses are zero (C12), as the Company does not incur its selling costs on the destroyed boomerangs,3 but G&A expense of $5 (B13) remains the same. Thus, there is a $110 net loss (C15).

Column D: The Difference Equals the Damages

Column D equals column B minus column C, and it equals the damages for lost inventory plus lost profits. BB loses $100 (D6 = B6 - C6) of sales. However, it still incurred the variable manufacturing costs and fixed overhead as before, i.e., there is no damage arising from those items (D8, D9, and D10 equal zero).

BB saves $10 (D12) of selling expenses, but there is no saving (D13) on G&A expense. Thus, the total damage is $90 (D15), which is $100 of lost sales minus $10 of selling expenses saved. This also equals -$20 -$110 = $90 (B15 - C15 = D15).

As mentioned earlier, column D is the total damages, while columns F and G allocate the total damage between the loss of inventory and lost profits.

Column F: Damages from the Lost Boomerangs

The value of the lost boomerang inventory is the amount of their direct manufacturing costs (direct materials, direct labor, and variable overhead), which totals $95 (F8, which equals B8 and C8).

Column G: Lost Profits

Our calculation of lost profits from lost sales begins with lost sales of $100 (G6, from D6). If BB had sold the boomerangs, it would have incurred the $95 (G8, which comes from B8 and C8) marginal cost of the variable manufacturing overhead. Note that G8 and F8 effectively cancel out each, as the $95 in F8 is a positive damage, while the $95 in G8 reduces lost profits and is a negative damage. Together they add to zero, although the accounting format slightly obscures that fact in showing G8 as a positive cost.

Fixed manufacturing overhead is not a component of damages, as it would have occurred with or without the fire. Thus, G9 equals zero, and total incremental costs of the lost sales are $95 (G8 + G9 = G10). Total incremental lost gross profit is $5 (G6 - G10 = G11). Note that the operational word is incremental, as accounting gross profit is -$5 (B11). Again, the difference between the lost profits calculation of gross profits lost and the accounting gross profits is the $10 of fixed costs, which are a legitimate expense in the original income statement in column B, but are not damages in columns D and G.

Selling expenses saved are $10 (G12). G&A expense, just like fixed manufacturing overhead, would have occurred with or without the fire, and thus also are not damages and therefore equal zero (G13).4 Total SG&A damages are $10 (G12 + G13 = G14), which leads to a lost profits calculation of -$5 (G15).

The sum of the lost boomerang inventory and lost profits calculations are $95 - $5 = $90 (F8 + G15 = D15).

Total Damage Formula on Destroyed Inventory

Now we can derive general formulas for both categories of damages. Rows 19 - 21 contain the first three equations, which are:

[1] Inventory Damage = Variable Mfg Costs = VM

Equation [1] merely states algebraically that which we have already discussed, which is that the inventory damage is the variable manufacturing costs. In my calculation of damages, I removed fixed manufacturing overhead from the plaintiff's calculation.

Equation [2] states that lost profits equal sales minus variable manufacturing costs minus selling expenses.

[2] Lost Profits = LP = S -VM - Sell Exp

Adding equations [1] and [2] gives us our formula for total damages on the destroyed inventory.

[3] Total Damage = S - Sell Exp Formula for Total Damages on Destroyed Inv

Note that the VM in equation [1] and the -VM in equation [2] cancelled out each other. This occurs for the reasoning mentioned above that the variable manufacturing costs are not part of total damages, as they would have occurred with or without the fire.

Cells B19 through B21 repeat the calculations above, which are equations [1] and [2], the separate components of the damage. Thus, Total Damages = VM of $95 (B19) + Lost Profits of -$5 (B20) = $90 (B21).

Next we test the total damage formula in equation [3]. Lost sales equal $90 (B25, from D6) and selling expenses saved equal $10 (B26, from D12), for a total damage of $90 (D27). Note that this equals B21 and confirms equation [3].

Scanty Information

One of the big challenges in the litigation out of which this article arose was that there was scanty financial statement information available to me. All financial information was provided in response to a demand and was provided to my attorney before I was engaged on the job. There was no ability to request additional financial information, and thus it was necessary to get creative and develop several different damage formulas.

With the little information that was available, calculating damages was like trying to put together a coherent picture puzzle when some of the pieces were missing, and it required that I calculate alternative formulas that would enable me to make the critical calculations with the data that were available.

Damage Formulas Based on Net Income

In the next section, we develop a damage formula based on net income. Equation [4] below is based on Table 1, B6 through B15. Equations [4] through [7] appear in rows 31 through 34 of Table 1. Equation [1] states that net income equals sales minus the sum of variable manufacturing costs, fixed manufacturing overhead, selling expenses, and G&A expense.

[4] NI = S - VM - FMOH - Sell Exp - G&A

Rearranging equation [4], we get:

[5] NI = (S - Sell Exp) - VM - FMOH- G&A

Note that the term in parentheses equals total damages, per equation [3]. Substituting equation [3] into equation [5], we get:

[6] NI = Damages - VM - FMOH- G&A

Rearranging the terms in equation [6], we get:

[7] Damages = NI + VM + FMOH + G&A Formula for Damages Based on NI

Let's try to understand the intuition behind equation [7]. Variable Manufacturing Costs (VM) are the damages from the lost inventory of boomerangs that were destroyed. The remaining three terms on the right-hand-side of equation [7] are the lost profits from the lost sale of the destroyed inventory.

We start with net income, which is a logical starting point to calculate lost profits. However, fixed manufacturing overhead (FMOH) was subtracted from sales in calculating net income, but it is not an incremental cost. Generally Accepted Accounting Principles (GAAP) requires using absorption costing for financial statements, which means subtracting fixed manufacturing overhead from net income. This is fine for presenting an income statement. However, absorption costing distorts the incremental profits analysis, which is what is relevant in damages calculations, and we have to convert to variable costing by adding back FMOH. Finally, G&A does not change with the fire, and it must be added back, as it is not a damage.

Cells B36 through B40 show the components of the damage calculation per equation [7], and again it adds to $90 (B40 = D15 = B21 = B27), as it should.

In the next series of equations, we will develop a formula for lost profits on the destroyed inventory based on net income. Note that these equations appear in rows 44 through 47.

We repeat equation [2] as equation [8], using LP for Lost Profits.

[8] LP = S -VM - Sell Exp

Next, we repeat equation [4] as equation [9].

[9] NI = S - VM - FMOH - Sell Exp - G&A

Subtracting equation [9] from equation [8], we get:

[10] LP - NI = FMOH + G&A

Rearranging the terms, we get:

[11] LP = NI + FMOH + G&A Formula for Lost Profits Based on Net Income

Cells B49 through B52 show the components of the damage calculations, and total lost profits equals -$5 (B52), which equals our calculations in G15. Thus, this calculation confirms equation [11].

Table 1B: Sample Damage Calculations with VM = $65

Table 1B is identical to Table 1A, except that cell B8 is $65 of variable manufacturing costs instead of $95 in Table 1. In this version, the Company is profitable. All other calculations flow through with the same logic. Note that total damages are still $90 (D15, B21, B27, and B40), even though gross profit and net income are positive. This demonstrates the accuracy of equations [3], [7], and [11].

Table 1B: Lost Profits Formulas Based on EBITDA for Lost Sales on Inventory Never Produced

It happened that Billabong's Boomerangs was a wholly-owned subsidiary of Sticks and Stones, Inc., a publicly traded firm. Most of the detailed financial information that we were given was for BB's local plant, and we had only summary financial statements for BB as a whole company. These summary statements were missing critical data that we needed for our damage calculations. Fortunately, we were able to find a few critical pieces of information about BB in the Sticks and Stones annual report, which is publicly available information. The most critical piece of information is that BB's EBITDA was -$40 million.

Table 1B repeats the data from Table 1A for the first 21 rows for columns B through D. However, there is a conceptual difference between Table 1A and Table 1B. In Table 1A, we are assuming the lost sales and the related lost profits are from the actual boomerangs that were burned, while Table 1B has a different twist to it. BB claimed that the fire caused an electrical outage, which caused the Company's nationwide computer system to go down. Therefore, its sales people at its national sales center did not have access to the system, and the Company could not effectively sell product for that day.5 Therefore, it claimed lost profits from lost sales for the entire company for one day. The accounting profits analysis of this type of lost sale is different than the lost profits on the destroyed inventory, as this inventory was never produced, and therefore the direct manufacturing were never incurred.

Thus, cells B6 through B15 and C6 are identical in Tables 1B and 1A. However, variable manufacturing costs are zero (C8) for the lost sales for the one day, which is different than the $65 in Table 1A. This change flows through C10 through C15, and net income is -$15 (C15), which is $65 higher than the net income of -$80 in Table 1A, C15. The $65 difference also flows through D8 through D15, as the difference in net income, i.e., the damage, equals $25 (D15), compared to $90 (Table 1A, D15).

We calculate the damage formula in rows 18 through 21, and the related equation appears immediately below.

[12] Lost Profits = S - VM - Sell Exp

Comparing this to equation [3], the difference is that in this damage formula, total lost profits are reduced by the variable manufacturing costs, which is not true of equation [3]. The reason for this is that for these lost sales, the inventory was never manufactured, as the plaintiff claimed that sales were made to order, and the inability to take orders for one day meant that the Company never produced the boomerangs. Therefore, the lost profits are lower here, since the Company never incurred the manufacturing costs-unlike the lost profits on the destroyed inventory.

Thus, lost profits equal the $100 of lost sales (B18, from D5), minus the sum of $65 (B19, from D7) of direct manufacturing costs saved by not making the inventory and $10 (B20, from D11) selling expenses saved, for a total damage of $25 (B21).

Next, we produce an equation for EBITDA (earnings before interest, taxes, depreciation and amortization) in equation [13].

[13] EBITDA = S - VM - FMOH - Sell Exp - G&A + Int + D

Subtracting equation [13] from equation [12], we get:

[14] LP - EBITDA = FMOH + G&A - Int - D

Adding EBITDA to both sides, we get:

[15] LP = EBITDA + FMOH + G&A - Int - D EBITDA Lost Profits Equation

The calculations of lost profits in B32 through B40 are based on equation [15].6 We were able to piece together the necessary information required in equation [15] from Billabong's Boomerangs local manufacturing plant's income statements, the Company's summary financial statement for the entire firm, and the data that were available as part of Sticks and Stones' annual statements. The lost profits for the year 2001, when the fire occurred, totaled -$20 million (B38), which represented a loss of $76,923 (B40) per working day.

Thus, the lost profits calculations of plaintiff's accountants of several hundreds of thousands of dollars were incorrect, and there were no positive lost profits for Billabong's Boomerangs.

When Reality May Vary With Our Assumptions

Contrary to our initial assumption, BB paid its employees for the day of the fire, even though they did not do productive work that day.7 That appears to "blow" our formulas for damage calculations.

The plaintiff, however, separately charged the defendant for the lost labor. Thus, it appears that we not counting the unproductive labor costs at all, and the plaintiff double counted it. The latter is true. In our case, it was easy enough to adjust the result by agreeing with-and thus, allowing-the plaintiff's separate charge for labor, leaving our formulas intact. This is true for both manufacturing labor, selling expenses, and for excess G&A necessitated by the fire, e.g., cleanup of the site.

The actual situation was even more complicated, as definitive information was lacking, and it appeared that there was some partial benefit from the "unproductive labor." However, the important point in this article is to calculate the damage formulas and illustrate the principles by which these formulas are correct. The practitioner may be called upon to make slight modifications to fit any deviation of the fact situation from the assumptions in the formulas.

Modification of Formulas for Wholesale and Retail Businesses

Obviously, wholesale and retail businesses do not have manufacturing costs. The only categories of expenses are cost of sales, fixed overhead (non-manufacturing, e.g., store rent), selling expenses, and G&A expenses. In the damage equations, cost of sales would take the place of variable manufacturing expense. The principles behind the formulas are the same as those in manufacturing. The actual formulas are simpler, as the accounting itself is simpler.

Summary

This article presents a series of formulas for calculating damages for the cost of destroyed inventory and for lost profits. For the latter, there are separate formulas when the lost sale relates to inventory that was produced and was not produced. This should provide a comprehensive, definitive framework for damage calculations. We also discussed how to modify these formulas for non-manufacturing companies and for deviations in the fact patterns.

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Jay Abrams, ASA, CPA, MBA, founder and head of Abrams Valuation Group (AVG), is one of those rare individuals who integrates theory and practice. He has valued businesses and consulted on mergers and acquisitions in a wide range of industries, provided valuations and discounts for fractional interests and restricted stock, and conducted independent statistical and mathematical research regarding problems facing businesses. During his 25 years of accounting and valuation experience, he has made, and continues to make, significant contributions to the science of valuing businesses. Mr. Abrams' book, Quantitative Business Valuation: A Mathematical Approach For Today's Professionals (McGraw-Hill, 2001) shows how to integrate advanced scientific methods into real-world valuation analysis.

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