# How do you compute the single-product breakeven?

The breakeven point is the sales volume (breakeven volume: $$Q_{be}$$) or sales revenues (breakeven revenues: $$R_{be}$$) at which 1) generated revenues equal total costs ($$R_{be} = TC_{be}$$), 2) contribution margin equals fixed costs ($$CM_{be} = FC_p$$) and thus 3) Operating Income equals zero ($$OI_{be} = 0$$).

The formulas for the breakeven volume ($$Q_{be}$$) and the breakeven revenues ($$R_{be}$$) can be derived from the profit equation introduced earlier.

Breakeven volume

\begin{aligned} OI_{be} = 0 & = Q_{be} \times UCM - FC_p \\ FC_p & = Q_{be} \times UCM \\ \frac{FC_p}{UCM} & = Q_{be} \\ & \leftrightarrow \\ Q_{be} & = \frac{FC_p}{UCM} \end{aligned}

Where $$FC_p$$ are the total fixed costs for the period $$p$$ and $$UCM$$ is the unit contribution margin, $$(P - V_C)$$. The economic interpretation of this equation is literally how many times should the company earn the unit conribution margin (so how many units should it sell) to cover its fixed costs?

Solution. First, compute the fixed costs for the period:

$FC_p = 1,500 + 850 + 4 \times 1,500 = 8,350$

Then, the unit contribution margin for each product:

\begin{aligned} UCM_{mojito} & = 7.5 - 2.55 = 4.95 \\ UCM_{cuba.libre} & = 7.5 - 3.05 = 4.45 \\ UCM_{cosmopolitan} & = 8 - 3.45 = 4.55 \end{aligned}

This means that every time I sell one mojito, my profit increases by 4.95 euros; every time I sell one cuba libre, my profit increases by 4.45 euros; and every time I sell one cosmopolitan, my profit increases by 4.55 euros. Finally I can compute how many times I should earn each unit contribution margin to cover my fixed costs:

\begin{aligned} Q_{bk}^{mojito} & = \frac{FC_p}{UCM_{mojito}} = \frac{8,350}{4.95} = 1,687 \quad mojitos \\ Q_{bk}^{cuba.libre} & = \frac{FC_p}{UCM_{cuba.libre}} = \frac{8,350}{4.45} = 1.877 \quad cuba \quad libres \\ Q_{bk}^{cosmopolitan} & = \frac{FC_p}{UCM_{cosmopolitan}} = \frac{8,350}{4.55} = 1.836 \quad cosmopolitans \end{aligned}

Note that I rounded up the break-even volumes.

Breakeven revenues

\begin{aligned} OI_{be} = 0 & = R_{be} \times CMR - FC_p \\ FC_p & = R_{be} \times CMR \\ \frac{FC_p}{CMR} & = R_{be} \\ & \leftrightarrow \\ R_{be} &= \frac{FC_p}{CMR} \end{aligned}

Where $$CMR$$ is the contribution margin ratio, $$\frac{(P-V_C)}{P} = \frac{(R_p - VC_p)}{R_p}$$. The economic interpretation of this equation is literally how many times should the company earn the contribution margin ratio (so how many euros of revenues should it earn) to cover its fixed costs?

Solution. You could get the break-even revenues by just multiplying the break-even volumes by the corresponding product price. Note that you get a “fictive” revenue if you use the non-rounded numbers because you can only sell complete units. This “fictive” revenue is what the second approach, based on the contribution margin ratio, reports:

\begin{aligned} CMR_{mojito} & = \frac{7.5 - 2.55}{7.5} = 66\% \\ CMR_{cuba.libre} & = \frac{7.5 - 3.05}{7.5} = 59.33\% \\ CMR_{cosmopolitan} & = \frac{8 - 3.45}{8} = 56.875\% \end{aligned}

This means that every euro of revenues from mojitos increase my profit by 0.66 cents; every euro of revenues from cuba libres increase my profit by 0.59 cents; and every euro of revenues from cosmopolitans increase my profit by 0.57 cents. And finally I can compute how many times I should earn each contribution margin ratio to cover my fixed costs:

\begin{aligned} R_{bk}^{mojito} & = \frac{FC_p}{CMR_{mojito}} = \frac{8,350}{0.66} = 12,651.52 \text{ euros from mojitos} \\ R_{bk}^{cuba.libre} & = \frac{FC_p}{CMR_{cuba.libre}} = \frac{8,350}{0.5933} = 14,073.03 \text{ euros from cuba libres} \\ R_{bk}^{cosmopolitan} & = \frac{FC_p}{CMR_{cosmopolitan}} = \frac{8,350}{0.56875} = 14,681.32 \text{ euros from cosmopolitans} \end{aligned}

Limitations of the break-even point

Since we are working with Operating Income (OI), both Earnings Before Taxes (EBT) and Net Income (NI) are negative if the company has debts and pays interests. This means that the wealth created by operations is not enough to remunerate investors. A more interesting indicator is therefore the Target Profit Point, which we will introduce later. We need before that to address the case of multiple products.

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