# How do you compute the multi-product break-even?

The approach for a multi-product break-even consists in synthesizing all products in a sort of blended product which is a weighted average of the others, using the sales share (in volume or in revenues) as weights. These shares represent a product mix, which we must assume to make computations.

The product mix refers to the relative proportions in which a company’s products are sold. It can be expressed in percentage of the volume or in percentage of the revenues.

If the product mix is stated in volumes, it is easier to work with volumes and unit contribution margins. If it is stated in revenues, it is easier to work with revenues and contribution margin ratios.

With a volume mix

When you have volume mix (relative proportions of each product in the total sales volume), you should compute a weighted average unit contribution margin (WUCM), i.e. the sum of the products’ unit contribution margins weighted by their share of the total volume sold (I insist on the volume!):

$WUCM = \sum_{d = 1}^{m} UCM^d \times \frac{Q^d}{\sum Q }$

where $$\frac{Q_d}{\sum Q}$$ is the percentage of the total sales volume due to the product $$d$$. One way to interpret the weighted average unit contribution margin is as the expected increase in contribution margin (and thus profit) for every additional unit sold, knowing that the probability of selling a specific product is assumed to be the share of the total volume that this product captures. Indeed, assume that you sell a unit, but you do not know from which product. Then, the product share in the volume mix is your best guess about the probability of earning the unit contribution margin of this particular product.

Solution. First, since the product mix is stated as percentages of the total volume, I compute my weighted average unit contribution margins in each case:

$WUCM_{mix1} = 4.95 \times 60\% + 4.45 \times 30\% + 4.55 \times 10\% = 4.76 \\ WUCM_{mix2} = 4.95 \times 30\% + 4.45 \times 30\% + 4.55 \times 40\% = 4.64$

The expected increase in profit for every additional unit sold is 4.76 euros for the first product mix and 4.64 euros with the second product mix. Here you can start to see the impact of changing a product mix when products do not have the same contribution. In a second step I can apply the same formulas as in the single product case:

\begin{aligned} Q_{bk}^{mix1} & = \frac{FC_p}{WUCM_{mix1}} = \frac{8,350}{4.76} = 1,755 \quad cocktails \\ Q_{bk}^{mix2} & = \frac{FC_p}{WUCM_{mix2}} = \frac{8,350}{4.6} = 1,800 \quad cocktails \end{aligned}

Just changing our product mix by putting more weight on a product having a slightly lower unit contribution margin increases our break-even and therefore, as we will see later, our operating risks. Note that you can infer the corresponding break-even revenues in two different ways:

• multiplying this total volume by the proportions of each product in the mix to get the volume of each product, which you then multiply by the corresponding price and sum to obtain total revenues;

• multiplying this total volume by a weighted average price (sum of each price weighted by the corresponding product share in the volume mix).

With a revenue mix

When you have revenue mix (relative proportions of each product in the total sales revenues), you should compute a weighted average contribution margin ratio (WCMR), i.e. the sum of the products’ contribution margin ratios weighted by their share of the total revenues (I insist on the revenues!):

$WCMR = \sum_{d = 1}^{m} CMR_d \times \frac{R_d}{\sum R }$

The weighted average contribution margin ratio is the expected increase in contribution margin for any additional euro of revenue, assuming that the probability of this additional euro of revenue coming from a specific product is equal to the share of this product in the total revenues.

Solution. First, since the product mix is stated as percentages of the total revenues, I compute my weighted average contribution margin ratios in each case:

$WUCM_{mix1} = \frac{4.95}{7.5} \times 60\% + \frac{4.45}{7.5} \times 30\% + \frac{4.55}{8} \times 10\% = 63\% \\ WUCM_{mix2} = \frac{4.95}{7.5} \times 30\% + \frac{4.45}{7.5} \times 30\% + \frac{4.55}{8} \times 40\% = 60.35\%$

The expected increase in profit for every additional euro of revenu is 0.63 cents for the first product mix and 0.6035 cents with the second product mix. Here again you can start to see the impact of changing a product mix when products do not have the same contribution. In a second step I can apply the same formulas as in the single product case:

\begin{aligned} R_{bk}^{mix1} & = \frac{FC_p}{WCMR_{mix1}} = \frac{8,350}{0.63} = 13,235.59 \text{ euros of revenues} \\ R_{bk}^{mix2} & = \frac{FC_p}{WCMR_{mix2}} = \frac{8,350}{0.6035} = 13,835.96 \text{ euros of revenues} \end{aligned}

Just changing our product mix by putting more weight on a product having a slightly lower contribution margin ratio increases our break-even and therefore, as we will see later, our operating risks.

In summary, the key to performing CVP analysis in a multi-product or service setting is to make an assumption about the relative mix of products or services sold. This mix can be stated in volumes or in revenues. If the product mix is stated in volumes, you MUST either compute a weighted average unit contribution margin or translate the volume mix into a revenues mix to compute a weighted average contribution margin ratio. If the product mix is stated in revenues, you MUST either compute a weighted average contribution margin ratio or translate the revenues mix into a volume mix to compute a weighted average unit contribution margin. Once the assumption about product mix has been made and the WUCM or WCMR computed, the CVP analysis can be conducted as if the company was selling a single composite product.