## 4. Multiple-product scenario in CVP analysis

In real life, companies produce a range of products, not just one kind as was assumed in our earlier examples. Different products will have different selling prices, variable costs per unit and, as a result, different contribution margins and contribution margin ratios.

Can CVP analysis work with this complication? The answer is yes. You just need to obtain some data about the product mix.

First, you need to know the proportion in which each of the products is sold. Then you can calculate the contribution margin for each product. After that you can define the weighted average contribution margin, which is used in the determination of the break-even point or the amount of sales required to gain a desired profit.

Let's illustrate this concept by using our Friends Company example. Assume that the company produces three (3) types of valves: truck valves, car valves, and motor-bike valves. The following data is available:

 Truck Valves Car Valves Motor-bike Valves (A) Share in physical volume sold, % 30% 45% 25% (B) Selling price per unit, \$ \$10 \$8 \$7 (C) Variable costs per unit, \$ \$7 \$6 \$5 (D) Contribution per unit, \$ (B - C) \$3 \$2 \$2 (E) Contribution margin ratio (D ÷ B) 0.30 0.25 0.29 (F) Fixed costs total, \$ \$10,000

To calculate the weighted average contribution margin you need to weight the contribution margin per unit of these three products and present it as "three-in-one":

Weighted Average Contribution Margin per Unit = 30% x \$3 + 45% x \$2 + 25% x \$2 = \$2.3

Now the break-even point may be calculated.

## 4.1. Contribution margin technique, break-even point in multiple-product companies

The calculation of the break-even point in a multiple-product company follows the same logic as in a single-product company. While the numerator will be the same fixed costs, the denominator will now be the weighted average contribution margin. The modified formula is as follows:

 Break-even Point (in units) = Fixed Costs Weighted Average Contribution Margin per Unit

For our example, the break-even point (in units) approximates 4,348 units (i.e., \$10,000 ÷ \$2.3). These 4,348 units are then allocated to different valve types according to the proportion defined in row 1 in the table above:

 Truck valves: 4,348 units x 30% = 1,304 units Car valves: 4,348 units x 45% = 1,957 units Motor-bike valves: 4,348 units x 25% = 1,087 units

So, Friends Company will break-even (i.e., will get neither profit nor loss) if it sells these volumes of valves at the given proportion of 30%:45%:25%.

It is important to note that changes in the product mix will result in different break-even points. For example, if the market situation changes and Friends Company switches to the product mix with proportions of 45%:30%:25%, the break-even point will change. In this case, the weighted average contribution margin per unit will be \$2.45 and zero profits will be earned when the unit sales equal around 4,082 valves.

The change occurred because the contribution ratio per unit of truck valves is the highest (\$3 per truck valve versus \$2 per car or motor-bike valve). Thus, more income can be generated by producing and selling truck valves and the break-even point is reached faster (with fewer total items produced and sold).

The break-even point for a multiple-product scenario can be calculated in dollars as well. Here also, the numerator is the same fixed costs. The denominator will now be weighted the average contribution margin ratio. The modified formula is as follows:

 Break-even Point (in dollars) = Fixed Costs Weighted Average Contribution Margin Ratio

Based on figures from the earlier table with information about the three valve types (see above), the break-even point will be reached at:

 Break-even Point (in dollars) = \$10,000 = \$36,364 30% x 0.3 + 45% x 0.25 + 25% x 0.29

Therefore, to achieve the break-even point, Friends Company has to sell valves for a total of \$36,364 (valid only when the proportion of sales is 30%:45%:25%).

Let's check the results: ∑ Contribution per Unit x Unit Sales - Fixed Costs, should approximately equal zero (rounding difference may arise):

\$3 x 1,304 + \$2 x 1,957 + \$2 x 1,087 - \$10,000 = 0

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