## Break-even analysis and profit-to-volume ratio

One of the fundamental ways of financial planning is related to the definition of probable profits at different levels of production and sales. To carry out the necessary calculations for this, the financiers apply the break-even analysis method and conduct its results further through the analysis of the profit-volume analysis.

Analysis of the break-even point is aimed at determining the volume of production and sales of products, in which the firm already pays for its costs, but does not yet make a profit. This is the break-even point. In this methodology, the fixed costs, variable costs and proceeds from sales are graphically or algebraically correlated.

## A graphical approach to break-even analysis

The analyst makes a coordinate system, on the vertical axis of which are located rubles, and on the horizontal - units of production (Figure 4.1). The following graphs are drawn in this system:

1. Fixed costs, which remain unchanged with increasing production volumes. This is expressed graphically in a straight line parallel to the horizontal axis of coordinates.

2. Variable costs are more or less directly proportional to the volume of production. They are usually represented as a straight line with a positive slope passing through the origin.

3. The total costs, equal to the sum of the costs of the constant and variable, change more or less directly in proportion to the change in production volumes and are therefore represented by an increasing straight line with a positive slope that starts at the point of intersection of the vertical axis with the schedule of fixed costs.

4. Proceeds from sales are considered directly proportional to the volume of production. This curve starts at the origin and (hopefully!) Goes steeper than the line of variable costs, yielding some revenue from each unit sold and crossing the line of total costs at the break-even point. Above this point, production is profitable, and below it is unprofitable.

* Fig. 4.1. * ** Basics of graphical break-even analysis **

The graphical analysis model described here works well for medium-scale production and stable conditions. In fact, the economic life of the firm is much more complicated, and linear dependencies, included in the simplest version of the breakeven analysis, sometimes require a theoretical and practical refinement. They bring to life a nonlinear model of breakeven analysis (Figure 4.2).

First of all, we note that the constant costs are not always simply constant. If the business of the company goes perfectly, and it increases production and sales, then there may come a time when the available production facilities will approach the level of their maximum technical capabilities. This will require the purchase of new equipment or the transition to a new technology. In the first case, the curve of constant costs will look like a step, and in the second, most likely, as an S-shaped.

* Fig. 4.2. * ** Basics of graphical non-linear breakeven analysis **

The second point of refinement of the model. It is customary to depict the variable cost curve as a straight line from the very beginning. This means that the costs of production of all units of production are the same. Actually this is not true. Production of the first units and even batches of products is often more costly for a company than for an established production. The reasons for this are inexperience of staff, lack of organization, small quantities of orders for raw materials (and, consequently, small discounts) and unrelated relations with suppliers, unstable sales channels and, accordingly, relatively high sales costs, etc. The cost price of a unit of production often increases also when approaching the maximum productivity of equipment or the need to increase administrative and managerial costs at really massive scale of production close to the technical maximum of the firm's productivity.

All this leads to the fact that the direct variable of costs from the direct to the 5-shaped, and the direct total costs in general takes a complex shape.

The graphic image of the proceeds from the sale of products is also complicated in the modern manufacturing service market environment. Very often the first sales with an unsettled system bring in less revenue, because you need to "pay discounts" for entering the market. Then the curve stabilizes its growth in the zone of optimal sales, and closer to the saturation of this market sector again reduces its growth due to the growing effort to push each new unit of products into a tight market. In other words, the income curve also becomes similar to the Latin letter "5". In the nonlinear model, a second break-even point may appear, after which further growth in production volumes becomes unprofitable. In this model, the problem of finding a break-even point is transformed into a search for a point of maximum profitability.

Although the nonlinear model is theoretically more accurate, a linear model is often used for practical calculations. The latter is accurate enough for cases in moderate loading of production capacity, relative stability of variable costs per unit of output and far enough from the starting stabilization zone and the limit load zone.

## Algebraic approach to breakeven analysis

The break-even point analysis can also be performed using mathematical formulas. Let's demonstrate two such techniques and examples of their application.

A. Calculation in terms of the number of units sold:

* U * =

*FС/Р - VC.*

where * U * - the number of units sold at the breakeven point; * FC * - fixed costs per unit of output; * P * - price per unit; * VC * - variable costs per unit of output.

** Example 1 **

Find the breakeven point of firm A if it has a fixed cost sum equal to 50,000 rubles, the price of a unit of production equal to 2 rubles, and variable costs of 1 ruble. per unit of output.

* Solution. *

* U * = 50000/(2 - 1) = 50,000 pieces of finished and sold products with sales of 100,000 rubles. ((50,000 x 2) = 100,000}.

B. Calculation in terms of ruble sales:

** S = FC/MS, **

where * S * is the sales volume in rubles at the breakeven point; * MS * - the marginal contribution as a share in the sales price, calculated by the formula

** (SP-VC)/PS. **

** Example 2 **

Find the breakeven point of firm B, if it has a fixed cost sum equal to 300,000 rubles. and a marginal contribution of 30%. Calculate break-even point in rubles and units of output.

* Solution. *

The sales volume in the breakeven point = 300 000/0.30 = 1 000000 rub.

The sales volume in production units can not be calculated, since the chain for the unit of production is unknown.

Modification of these formulas allows you to analyze the relationship of profit - the volume of production. We leave the break-even point in the zone of profitable operations and, accordingly, rewrite the formulas. As a result, we get

** S = (FC + P)/MS, **

where * S * is the volume of sales in rubles required to achieve the level of profit * P; *

** U ** =

*(FC + P)/(SP-VC).*

where * U * is the sales volume in the product units necessary to achieve the profit level * P. *

These formulas can be applied to different indicators of profit - before taxes and interest, before taxes, after taxes and interest. The corresponding indicator of profitability is meant by * P. *

We will illustrate the application of these formulas by examples. Consider a firm in which the selling price per unit of production is 6 rubles, with variable costs of 4 rubles, fixed costs of 100,000 rubles, borrowed funds in the amount of 300,000 rubles. iodine 10% per annum and a 28% tax rate.

** Example 1 **

If the company seeks to get 200,000 rubles. profit before interest and taxes, then what level of sales (5) should it achieve?

* Answer: *

* S * = (100,000 + 200,000)/(6 - 4)/6 = 300,000/0.335 = 900,000 rubles.

** Example 2 **

If the firm aspires to receive 250 000 rbl. profit before taxes, then what level of sales (* S *) should it achieve?

* Responses. *

5 = (100,000 + 30000 + 250000)/0.335 = 1 140,000 rubles.

* U = * 380000/(6 - 4) = 190,000 units of products.

** Example 3 **

If the company seeks to get 300,000 rubles. net profit, then what level of sales (* S *) should it achieve?

* Responses. *

Profit before taxes = 300000/(1 - 0.28) = 416667 rubles.

5 = (100000 + 30000 + 416667)/0.335 = 546667/0.335 = 1 640000 rub.

* U = * 546667/(6 - 4) = 273333 units of production.

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