BUSINESS MATHEMATICS

1. BUSINESS MATHEMATICS Breakeven Analysis

2. Breakeven Analysis Defined • Breakeven can be defined as the minimum level of sales revenue or units of goods that an organisation must meet or produce to cover all costs. • To determine these level, costs must be determined appropriately and accurately. • To determine cost accurately, the cost of every factor of production (land, labour, capital, and entrepreneur must be accounted for.

3. Break-Even Analysis • The Objective of breakeven analysis is to find the level of output at which cost equals revenue • Requires estimation of fixed costs, variable costs, and revenue

4. Break-Even Analysis • Fixed costs are costs that continue even if no units are produced • Depreciation, taxes, debt, mortgage payments • Variable costs are costs that vary with the volume of units produced • Labor, materials, portion of utilities • Contribution is the difference between selling price and variable cost

5. Break-Even Analysis Assumptions • Costs and revenue are linear functions • Generally not the case in the real world • We actually know all the costs • Very difficult to accomplish • There is no time value of money

6. DDecision making and Breakeven Analysis • How many units must be sold to breakeven? • How many units must be sold to achieve a target profit? • How will profits and or breakeven volume be affected if fixed cost, variable cost, and or total revenue increased or reduced?

7. Key Terminology: Breakeven Analysis • Break even point-the point at which a company makes neither a profit or a loss (i.e. total revenue is equal to total cost – TR = TC) • Contribution per unit-the sales price minus the variable cost per unit. It measures the contribution made by each item of output to the fixed costs and profit of the organisation. (i.e. price per unit – variable cost per unit)

8. Key Terminology ctd. • Margin of safety-a measure in which the budgeted volume of sales is compared with the volume of sales required to break even. (i.e. Budgeted volume of sales – Breakeven volume) Example: If a firm budgets to sell 2000 units and it requires 1600 units to breakeven, then the margin of safety is: 2000 – 1600 = 400. (We could also use the value of these goods in cedis to determine the margin of safety) • Marginal Cost – cost of producing one extra unit of output

9. Breakeven Formula • Breakeven volume can be determined mathematically or graphically. Fixed Cost Price per unit – Variable cost per unit Selling Price per unit – Variable Cost per unit = Contribution per unit NB: This formula is used to determine the breakeven for single items only. To determine breakeven for multiple items, the multiple case formula is used.

10. BEPq = Break-even point in units P= Price per unit (after all discounts) q = Number of units produced TR = Total revenue = P x q F = Fixed costs Vq = Variable costs per unit TC = Total costs = F + Vx F P - V BEPq = Break-Even Analysis Break-even occurs when TR = TC or Pq = F + Vq

11. Example • Ante Araba sells biscuits in a shop around the corner. If her monthly fixed cost is Ȼ1,800 and her variable cost per unit is Ȼ6.75, determine the number of biscuits Ante Araba must sell to breakeven if a biscuit sells at Ȼ9.25. • Solution BEPq = f = 1800 =Ȼ 654.5 p – v 9.25 – 6.75

12. BEPq = = F p - v $10,000 [400 – (1.50 + 0.75)] Example Determine the breakeven volume if Fixed costs = Ȼ10,000 Material = Ȼ0.75/unit Direct labor = Ȼ1.50/unit Selling price = Ȼ4.00 per unit Note that variable cost was given here as cost of labour and material.

13. BEPq = Break-even point in units BEPȻ = Break-even point in dollars P = Price per unit (after all discounts) q = Number of units produced TR = Total revenue = PQ F = Fixed costs V = Variable costs TC = Total costs = F + Vq BEP$ = BEPq ×P = P = = F P - V F 1 - V/P F (P - V)/P Break-Even in Monetary Terms Profit = TR - TC = Pq- (F + Vq) = Pq- F - Vq = (P - V)q - F

14. BEPȻ= = F 1 - (V/P) = = Ȼ22,857.14 BEPq= = = 5,714 units Ȼ10,000 1 - [(1.50 + .75)/(4.00)] Ȼ10,000 .4375 Ȼ10,000 4.00 - (1.50 + .75) F P - V Example Determine the value of sales at breakeven if: Fixed costs = Ȼ10,000 Material = Ȼ0.75/unit Direct labor = Ȼ1.50/unit Selling price = Ȼ4.00 per unit

15. 50,000 – 40,000 – 30,000 – 20,000 – 10,000 – – Total Revenue Break-even point Total costs Cedis Fixed costs | | | | | | 0 2,000 4,000 6,000 8,000 10,000 Units Graphical Representation of Example in Previous Slide

16. – 900 – 800 – 700 – 600 – 500 – 400 – 300 – 200 – 100 – – Total revenue line Total cost line Break-even point Total cost = Total revenue Profit corridor Cost/Revenue in cedis Variable cost Loss corridor Fixed cost | | | | | | | | | | | | 0 100 200 300 400 500 600 700 800 900 1000 1100 Volume (units per period) Graphical Breakeven Analysis

17. Multiproduct Breakeven Analysis • What we have discussed so far applies to single products or items only. Where the firm deal with multiple products, the multiple product formula is used. • The breakeven under the multiple product is determine in monetary terms only. • That means that if the firm wants to know how much of each item to sell to breakeven, they have to use the single case approach

18. F BEPȻ= ∑1 - x (Wi) Vi Pi Multiple Product Formula Multiproduct Case where V = variable cost per unit P = price per unit F = fixed costs W = percent each product is of total dollar sales i = each product

19. Annual Forecasted Item Price Cost Sales Units SardineȻ2.95Ȼ1.25 7,000 Soft drink .80 .30 7,000 Biscuit 1.55 .47 5,000 Pencil .75 .25 5,000 Milk 2.85 1.00 3,000 Multiproduct Example Fixed costs = Ȼ3,500 per month

20. Annual Forecasted Item Price Cost Sales Units Sandwich $2.95 $1.25 7,000 Soft drink .80 .30 7,000 Baked potato 1.55 .47 5,000 Tea .75 .25 5,000 Salad bar 2.85 1.00 3,000 Annual Weighted Selling Variable Forecasted % of Contribution Item (i) Price (P) Cost (V) (V/P) 1 - (V/P) Sales $ Sales (col 5 x col 7) Sardine $2.95 $1.25 .42 .58 $20,650 .446 .259 Soft drink .80 .30 .38 .62 5,600 .121 .075 Biscuits 1.55 .47 .30 .70 7,750 .167 .117 Pencil .75 .25 .33 .67 3,750 .081 .054 Milk 2.85 1.00 .35 .65 8,550 .185 .120 $46,300 1.000 .625 Multiproduct Example Fixed costs = $3,500 per month

21. F BEPȻ = ∑1 - x(Wi) Vi Pi Ȼ3,500 x 12 .625 = = Ȼ67,200 Multiproduct Example Fixed costs = $3,500 per month and so to get annual, we multiply by 12

22. Margin of Safety • The difference between budgeted or actual sales and the breakeven point • The margin of safety may be expressed in units or revenue terms • Shows the amount by which sales can drop before a loss will be incurred

23. Example 1 Using the following data, calculate the breakeven point, contribution per unit and margin of safety in units: • Selling Price = Ȼ50 • Variable Cost = Ȼ40 • Fixed Cost = Ȼ70,000 • Budgeted Sales = 7,500 units

24. Example 1: Solution • Contribution = Ȼ50 - Ȼ40 = Ȼ10 per unit • Breakeven point = Ȼ70,000/Ȼ10 = 7,000 units • Margin of safety = 7500 – 7000 = 500 units • To determine the margin of safety in monetary terms, we multiple 500 by 50 to get Ȼ25,000

25. Target Profits • Sometime, firms do not just want to breakeven. They may want a target profit • In that case, contribution per unit will need to cover profit as well as fixed costs • Required profit is accordingly treated as an addition to Fixed Costs

26. Example 2 Using the following data, calculate the level of sales required to generate a profit of Ȼ10,000: • Selling Price = Ȼ35 • Variable Cost = Ȼ20 • Fixed Costs = Ȼ50,000

27. Example 2: Solution • Contribution = Ȼ35 – Ȼ20 = Ȼ15 • Level of sales required to generate profit of Ȼ10,000: Ȼ50,000 + Ȼ10,000 Ȼ15 = 4000 units • The firm will need to sell 4000 units tonot only breakeven, but make a target profit of Ȼ10,000